Adjoint(x) is the linear transformation mapping g to g defined by Adjoint(x)(y) = [x, y] for all y in g.  The linear transformation Adjoint(x) always defines a derivation on g.

The linear transformation AdjointExp(x) is the Lie algebra isomorphism defined by AdjointExp(x) = exp(Adjoint(x)) of the vector x in g.

Adjoint() returns the list of adjoint matrices for the basis vectors of the current algebra g.

Adjoint(alg) returns the list of adjoint matrices for the basis vectors of the algebra alg.

Adjoint(x, h) calculates the restriction of Adjoint(x) to the subspace h (h must be an Adjoint(x) invariant subspace).

Adjoint(x, h, k) calculates Adjoint(x) on the vector space quotient g/k with respect to the basis determined by h (k must be an Adjoint(x) invariant subspace).

The commands Adjoint and AdjointExp are part of the DifferentialGeometry:-LieAlgebras package.  They can be used in the form Adjoint(...) and AdjointExp(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Adjoint(...) and DifferentialGeometry:-LieAlgebras:-AdjointExp(...).