Trace forms of solvable Lie algebras annihilate the derived algebra ($B_V([\mathfrak{g},\mathfrak{g}],\mathfrak{g})=0$)

For a finite-dimensional complex representation of a solvable Lie algebra, the Lie theorem gives a basis in which all representing matrices are upper triangular. Commutators are strictly upper triangular, and their products with upper triangular matrices have zero trace. Thus the derived algebra lies in the radical of a bilinear form. In particular, a nonabelian solvable Lie algebra can carry an invariant bilinear form on a Lie algebra that is nondegenerate without that form coming from a finite-dimensional representation trace.

// Target: semisimple-lie-algebra.bigb