Example 1.

The Lie algebra of 4 x 4 Upper triangular matrices is a 10 dimensional Lie algebra which is naturally graded - g0 consists of the matrices with only non-zero elements on the diagonal, g1 consists of the matrices with non-elements immediately above the diagonal (the super diagonal) and so on.

We use Query to verify this.  First we use the program MatrixAlgebras to generate the Lie algebra data structure for the Lie algebra of upper triangular matrices.  Here eij denotes the matrix with a 1 in the i-th row and j-th column.

To display the Lie algebra multiplication table, we need to increase the value of the interface parameter rtablesize.

Now define the 4 subspaces which will define our gradation.

DifferentialGeometry, LieAlgebras, MatrixAlgebras, MultiplicationTable, Query