Is there a name for this kind of space? Assume a Riemannian symmetric space $G/H$ where the decomposition of the Lie algebra of $G$ is $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$. It is a known fact that if $\mathfrak{h}$ is the Lie algebra of $H$ then
$$
[\mathfrak{h},\mathfrak{h}]\subset \mathfrak{h}], \quad [\mathfrak{h}, \mathfrak{m}] \subset \mathfrak{m},\quad [\mathfrak{m},\mathfrak{m}] \subset \mathfrak{h}.
$$
Is there a special name to call the spaces $G/H$ where the stronger condition
$$
[\mathfrak{m},\mathfrak{m}] = \mathfrak{h}
$$
holds? Or a name for such decomposition?

The decomposition with the stronger condition is called effective. It appears in connection with
pseudo-Riemannian symmetric spaces, where one deals with a special case of the symmetric decomposition $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{m}$, namely that the decomposition is effective, i.e., that it is minimal, which precisely means that $[\mathfrak{m},\mathfrak{m}] = \mathfrak{h}$, and that $\mathfrak{h}$ does not contain a non-trivial ideal of $\mathfrak{g}$. For details see here.
