= Root bracket and the opposite-root exception
{title2=$[\mathfrak g_\alpha,\mathfrak g_{-\alpha}]\subset\mathfrak h$}
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 $\alpha+\beta\ne0$. 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.
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