Proposition 1

(i) The subalgebras of $\mathfrak{g}$ stable under ad$\mathfrak{h}$ are the vector subspaces of the form $\mathfrak{h}'+\mathfrak{g}^P$, where P is a closed subset of R and $\mathfrak{h}'$ is a vector subspace of $\mathfrak{h}$ containing $\mathfrak{h}_{P\cap(-P)}$.

(ii) Let $\mathfrak{h}',\mathfrak{h}''$ be vector subspaces of $\mathfrak{h}$ and $P,Q$ closed subsets of R, with $\mathfrak{h}'\supset \mathfrak{h}_{P\cap(-P)},\mathfrak{h}''\subset \mathfrak{h}'$ and $Q\subset P$. Then $\mathfrak{h}''+\mathfrak{g}^Q$ is an ideal of $\mathfrak{h}'+\mathfrak{g}^P$ if and only if

$(P + Q)\cap R\subset Q$ and $\mathfrak{h}_{P\cap(-Q)}\subset \mathfrak{h}''\subset \bigcap_{\alpha\in P,\alpha \notin Q}$ Ker $\alpha$.

Assertion (i) follows immediately from Lemmas 1 and 2. Let $\mathfrak{h}',\mathfrak{h}'',P,Q$ be as in (ii). Then

Hence, $\mathfrak{h}''+\mathfrak{g}^Q$ is an ideal of $\mathfrak{h}'+\mathfrak{g}^P$ if and only if

This implies (ii).