Example 1.

In this example we illustrate the fact that the result of Inquiry("Indecomposable") does not depend upon the choice of basis for the Lie algebra.

First we initialize a Lie algebra.

Now we make a change of basis in the Lie algebra.  In this basis it is not possible to see that the Lie algebra is decomposable by examining the multiplication table.

Both Alg1 and Alg2 are seen to be decomposable.

Example 2

Here is the simplest example of a solvable Lie algebra which is absolutely decomposable but not decomposable.

First we initialize the Lie algebra and display the multiplication table.

The algebra is indecomposable over the real numbers.

The algebra is decomposable over the complex numbers.

The explicit decomposition of this Lie algebra is given in the help page for the command Decompose.

DifferentialGeometry, LieAlgebras, DecomposeAlgebra, LieAlgebraData, Query