 Two-root decomposition with a nondegenerate trace form

For a finite-dimensional complex Lie algebra, let a maximal torus $\mathfrak{t}$ give a decomposition $\mathfrak{g}={\mathfrak{g}}_{\alpha }\oplus \mathfrak{t}\oplus {\mathfrak{g}}_{\beta }$. If $\mathfrak{g}$ has a nondegenerate trace form, then $\beta =-\alpha$, the opposite root spaces generate ${\mathfrak{sl}}_2$, andwhere $\ker \alpha \subset \mathfrak{t}$ is central and abelian.