= Triangular decomposition of a Lie algebra
{title2=$\mathfrak g=\mathfrak n^-\oplus\mathfrak h\oplus\mathfrak n^+$}
Choosing a <positive system of a root system> for a complex <semisimple Lie algebra> divides its <root spaces> into negative and positive ones. Their respective sums $\mathfrak n^-$ and $\mathfrak n^+$, together with the <Cartan subalgebra> $\mathfrak h$, give this <direct sum> of <vector spaces>. Both $\mathfrak n^\pm$ are <nilpotent Lie algebras>, and $\mathfrak h\oplus\mathfrak n^+$ is a <Borel subalgebra>. The <Poincare-Birkhoff-Witt theorem> turns this into an ordered vector-space decomposition of the <universal enveloping algebra>, useful for building <Verma modules>.
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