Derivations - Maple Help

LieAlgebras[Derivations] - find the derivations of a Lie algebra, find the derivations of a general non-commutative algebra

Calling Sequences

Derivations(Algname, "keyword")

Parameters

Algname   - (optional) name or string, the name of a Lie algebra

keyword   - one of the 3 keywords "Inner", "Full", or "Outer"

Description

 • Let be a $n$-dimensional Lie algebra. An matrix $B$ is a derivation forif the associated linear transformation mapping satisfies

for all .

The set of all derivations defines a matrix Lie algebra denoted by Der$\left(\mathrm{𝔤}\right)$. For each the adjoint matrix adis a derivation -- these are the inner derivations InnDer($\mathrm{𝔤})$. The inner derivations define an ideal in Der($\mathrm{𝔤})$and the quotient Lie algebra Der($\mathrm{𝔤}$)/InnDer($\mathrm{𝔤})$ is the Lie algebra of outer derivations.

 • Let $\mathrm{𝔸}$ be a $n$-dimensional Lie algebra (such as the octonion, a Jordan algebra, or a Clifford algebra. See AlgebraLibraryData). An matrix is a derivation for $\mathrm{𝔸}$ if the associated linear transformation mapping satisfies

for all .

 • Derivations(Algname, "Inner") returns a list of linearly independent matrices which defines a basis for the Lie algebra of inner derivations for the Lie algebra Algname.
 • Derivations(Algname) or Derivations(Algname, "Full") returns a list of linearly independent matrices which defines a basis for the Lie algebra of all derivations for the Lie algebra Algname.
 • Derivations(Algname, "Outer") returns a list of linearly independent matrices which gives a representative list of the outer derivations for the Lie algebra Algname.
 • If Algname is a general non-commutative algebra, then Derivations(Algname) computes the derivations of this algebra.
 • The command Derivations is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Derivations(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Derivations(...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

First initialize a Lie algebra and display the Lie bracket multiplication table.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[4\right]\right],\left[\left[\left[1,4,1\right],1\right],\left[\left[2,4,1\right],1\right],\left[\left[2,4,2\right],1\right],\left[\left[3,4,3\right],1\right]\right]\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$
 Alg1 > $\mathrm{MultiplicationTable}\left("LieBracket"\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (2.1)

For the Lie algebra Alg1 we find that Derivations(Alg1, "Inner") is 4 dimensional and Derivations(Alg1) is 8 dimensional.

 Alg1 > $\mathrm{Inner}≔\mathrm{Derivations}\left("Inner"\right)$
 ${\mathrm{Inner}}{:=}\left[\left[\begin{array}{rrrr}{1}& {1}& {0}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]$ (2.2)
 Alg1 > $\mathrm{Der}≔\mathrm{Derivations}\left("Full"\right)$
 ${\mathrm{Der}}{:=}\left[\left[\begin{array}{rrrr}{1}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]$ (2.3)
 Alg1 > $\mathrm{Outer}≔\mathrm{Derivations}\left("Outer"\right)$
 ${\mathrm{Outer}}{:=}\left[\left[\begin{array}{rrrr}{1}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]$ (2.4)

We can study the properties of Derivations(Alg1) by initializing these matrices as a Lie algebra. We use as a basis for Derivations(Alg1) the inner and outer derivations.

 Alg1 > $\mathrm{Basis}≔\left[\mathrm{op}\left(\mathrm{Inner}\right),\mathrm{op}\left(\mathrm{Outer}\right)\right]:$
 Alg1 > $\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{Basis},\mathrm{DerAlg}\right)$
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e6}}\right]$ (2.5)
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L2},\left[E\right],\left[a\right]\right):$

We see that the derivation algebra is solvable.

 DerAlg > $\mathrm{Query}\left("Solvable"\right)$
 ${\mathrm{true}}$ (2.6)

We check that the span of the vectors (corresponding to the inner derivations) define an ideal.

 DerAlg > $\mathrm{Query}\left(\left[\mathrm{E1},\mathrm{E2},\mathrm{E3},\mathrm{E4}\right],"Ideal"\right)$
 ${\mathrm{true}}$ (2.7)

We compute the quotient algebra of outer derivations.

 DerAlg > $\mathrm{L3}≔\mathrm{QuotientAlgebra}\left(\left[\mathrm{E1},\mathrm{E2},\mathrm{E3},\mathrm{E4}\right],\left[\mathrm{E5},\mathrm{E6},\mathrm{E7},\mathrm{E8}\right],\mathrm{OuterAlg}\right)$
 ${\mathrm{L3}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$ (2.8)
 DerAlg > $\mathrm{DGsetup}\left(\mathrm{L3}\right)$
 ${\mathrm{Lie algebra: OuterAlg}}$ (2.9)

Example 2.

We show that the derivations of the octonions form a 14-dimensional semi-simple Lie algebra (which can be seen to be compact real form of the exceptional Lie algebra ${g}_{2}$).

 > $\mathrm{L4}≔\mathrm{AlgebraLibraryData}\left("Octonions",\mathrm{Oct}\right)$
 ${\mathrm{L4}}{:=}\left[{{\mathrm{e1}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e1}}{.}{\mathrm{e2}}{=}{\mathrm{e2}}{,}{\mathrm{e1}}{.}{\mathrm{e3}}{=}{\mathrm{e3}}{,}{\mathrm{e1}}{.}{\mathrm{e4}}{=}{\mathrm{e4}}{,}{\mathrm{e1}}{.}{\mathrm{e5}}{=}{\mathrm{e5}}{,}{\mathrm{e1}}{.}{\mathrm{e6}}{=}{\mathrm{e6}}{,}{\mathrm{e1}}{.}{\mathrm{e7}}{=}{\mathrm{e7}}{,}{\mathrm{e1}}{.}{\mathrm{e8}}{=}{\mathrm{e8}}{,}{\mathrm{e2}}{.}{\mathrm{e1}}{=}{\mathrm{e2}}{,}{{\mathrm{e2}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e2}}{.}{\mathrm{e5}}{=}{\mathrm{e6}}{,}{\mathrm{e2}}{.}{\mathrm{e6}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e2}}{.}{\mathrm{e7}}{=}{-}{\mathrm{e8}}{,}{\mathrm{e2}}{.}{\mathrm{e8}}{=}{\mathrm{e7}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e4}}{,}{{\mathrm{e3}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e3}}{.}{\mathrm{e4}}{=}{\mathrm{e2}}{,}{\mathrm{e3}}{.}{\mathrm{e5}}{=}{\mathrm{e7}}{,}{\mathrm{e3}}{.}{\mathrm{e6}}{=}{\mathrm{e8}}{,}{\mathrm{e3}}{.}{\mathrm{e7}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e3}}{.}{\mathrm{e8}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e4}}{.}{\mathrm{e1}}{=}{\mathrm{e4}}{,}{\mathrm{e4}}{.}{\mathrm{e2}}{=}{\mathrm{e3}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e4}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e4}}{.}{\mathrm{e5}}{=}{\mathrm{e8}}{,}{\mathrm{e4}}{.}{\mathrm{e6}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e4}}{.}{\mathrm{e7}}{=}{\mathrm{e6}}{,}{\mathrm{e4}}{.}{\mathrm{e8}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e5}}{.}{\mathrm{e1}}{=}{\mathrm{e5}}{,}{\mathrm{e5}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e5}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e5}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e8}}{,}{{\mathrm{e5}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e5}}{.}{\mathrm{e6}}{=}{\mathrm{e2}}{,}{\mathrm{e5}}{.}{\mathrm{e7}}{=}{\mathrm{e3}}{,}{\mathrm{e5}}{.}{\mathrm{e8}}{=}{\mathrm{e4}}{,}{\mathrm{e6}}{.}{\mathrm{e1}}{=}{\mathrm{e6}}{,}{\mathrm{e6}}{.}{\mathrm{e2}}{=}{\mathrm{e5}}{,}{\mathrm{e6}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e8}}{,}{\mathrm{e6}}{.}{\mathrm{e4}}{=}{\mathrm{e7}}{,}{\mathrm{e6}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e6}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e6}}{.}{\mathrm{e7}}{=}{-}{\mathrm{e4}}{,}{\mathrm{e6}}{.}{\mathrm{e8}}{=}{\mathrm{e3}}{,}{\mathrm{e7}}{.}{\mathrm{e1}}{=}{\mathrm{e7}}{,}{\mathrm{e7}}{.}{\mathrm{e2}}{=}{\mathrm{e8}}{,}{\mathrm{e7}}{.}{\mathrm{e3}}{=}{\mathrm{e5}}{,}{\mathrm{e7}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e7}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e7}}{.}{\mathrm{e6}}{=}{\mathrm{e4}}{,}{{\mathrm{e7}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e7}}{.}{\mathrm{e8}}{=}{-}{\mathrm{e2}}{,}{\mathrm{e8}}{.}{\mathrm{e1}}{=}{\mathrm{e8}}{,}{\mathrm{e8}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e8}}{.}{\mathrm{e3}}{=}{\mathrm{e6}}{,}{\mathrm{e8}}{.}{\mathrm{e4}}{=}{\mathrm{e5}}{,}{\mathrm{e8}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e4}}{,}{\mathrm{e8}}{.}{\mathrm{e6}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e8}}{.}{\mathrm{e7}}{=}{\mathrm{e2}}{,}{{\mathrm{e8}}}^{{2}}{=}{-}{\mathrm{e1}}\right]$ (2.10)
 > $\mathrm{DGsetup}\left(\mathrm{L4}\right):$

We find that the derivation algebra is 14-dimensional

 > $\mathrm{Der}≔\mathrm{Derivations}\left(\mathrm{Oct}\right)$
 ${\mathrm{Der}}{:=}\left[\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {-}{1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-}{1}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-}{1}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {-}{1}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}& {-}{1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{1}& {0}& {0}& {0}\end{array}\right]\right]$ (2.11)
 > $\mathrm{nops}\left(\mathrm{Der}\right)$
 ${14}$ (2.12)

Calculate the structure equations for the derivations, initialize ,and check that the derivation algebra is semi-simple.

 Oct > $\mathrm{L5}≔\mathrm{LieAlgebraData}\left(\mathrm{Der},\mathrm{Alg5}\right)$
 ${\mathrm{L5}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e5}}{-}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}{-}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e4}}{-}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e5}}{+}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e3}}{-}{\mathrm{e11}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e5}}{+}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e2}}{-}{2}{}{\mathrm{e13}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e13}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e1}}{+}{\mathrm{e14}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e6}}{-}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e7}}{-}{\mathrm{e12}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e1}}{+}{\mathrm{e14}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e2}}{-}{\mathrm{e13}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e5}}{-}{\mathrm{e9}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e7}}{-}{\mathrm{e12}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{-}{2}{}{\mathrm{e11}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e11}}\right]{=}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e7}}{-}{2}{}{\mathrm{e12}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e11}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e13}}\right]{=}{-}{2}{}{\mathrm{e14}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e14}}\right]{=}{2}{}{\mathrm{e13}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e14}}\right]{=}{-}{2}{}{\mathrm{e12}}\right]$ (2.13)
 Oct > $\mathrm{DGsetup}\left(\mathrm{L5}\right)$
 ${\mathrm{Lie algebra: Alg5}}$ (2.14)
 Oct > $\mathrm{Query}\left("Semisimple"\right)$
 ${\mathrm{true}}$ (2.15)