Remark

It follows from Lemma 5 that every ideal of $\mathfrak{a}$ is homogeneous, since it is stable under the ad $h_{\alpha}$.

Denote by $Q_+$ (resp. $Q_-$) the set of linear combinations of elements of B with positive (resp. negative) integer coefficients, not all zero. Put $\mathfrak{a}_+=$ $\sum_{\mu\in Q_+}a^\mu$ and $\mathfrak{a}_-=\sum_{\mu\in Q_-}\mathfrak{a}^\mu$. Since $Q_++ Q_+\subset Q_+$ and $Q_-+ Q_-\subset Q_-,\mathfrak{a}_+$

and $\mathfrak{a}_-$ are Lie subalgebras of $\mathfrak{a}$.