Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 102 / 2 / ii / Solution

A torus $\mathfrak{t}$ in a Lie algebra is an abelian subalgebra whose elements act semisimply in the Adjoint representation. Its weight-space decomposition is

Invariance of the nondegenerate Trace form of a Lie algebra representation givesNondegeneracy therefore forces $\beta =-\alpha$. The standard sl2 subalgebra associated with a root argument gives one-dimensional opposite root spaces with vectors $e\in {\mathfrak{g}}_{\alpha }$, $f\in {\mathfrak{g}}_{-\alpha }$ and $h_{\alpha }\in \mathfrak{t}$ satisfyingTheir span is a copy of ${\mathfrak{sl}}_2$.

SetThen $\dim \mathfrak{h}=\dim \mathfrak{t}-1$, $\mathfrak{t}=\mathbb{C}{h}_{\alpha }\oplus \mathfrak{h}$, and every element of $\mathfrak{h}$ commutes with $e$, $f$, and $\mathfrak{t}$. Hence $\mathfrak{h}$ is abelian andas a direct sum of commuting Lie algebras. This is the two-root decomposition with a nondegenerate trace form.