Root space (${\mathfrak{g}}_{\alpha }$)

For a root $\alpha$, its root space is the simultaneous eigenspaceof the Adjoint representation of the Cartan subalgebra.

For root spaces ${\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }$, the Jacobi identity gives $[{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }]\subset {\mathfrak{g}}_{\alpha +\beta }$ when $\alpha +\beta \ne 0$. If the bracket is nonzero, that nonzero weight must be a root. When $\beta =-\alpha$, the bracket instead lies in the Cartan subalgebra and can be nonzero; zero is not a root. Thus the common nonzero-bracket implication requires the opposite-root exception.

A root vector is a nonzero element of a root space ${\mathfrak{g}}_{\alpha }$ in the root-space decomposition of a semisimple Lie algebra.