 Killing form of a nilpotent Lie algebra vanishes

Every Adjoint representation operator raises the lower central series of a Lie algebra filtration. In a basis adapted to that finite filtration, all adjoint operators are strictly triangular. Their products have zero trace, so the Killing form vanishes. Consequently its radical of the Killing form is the whole algebra. For strictly upper triangular matrices the filtration can be seen directly by the gap between the column and row indices.

// Target: algebra.bigb