= Trace forms of solvable Lie algebras annihilate the derived algebra
{title2=$B_V([\mathfrak g,\mathfrak g],\mathfrak g)=0$}
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
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