Remark

— 1) The mutually inverse bijections $K \mapsto g(K)$ and $\mathfrak{h} \mapsto I(\mathfrak{h})$ are inclusion reversing; therefore $\mathfrak{h} \mapsto I(\mathfrak{h})$ is an isomorphism of the ordered set $\mathscr{L}$ onto the ordered set opposite to $\mathscr{C}$. Hence we obtain the relation $I(\mathfrak{h} \cap \mathfrak{h}') = E^p(I(\mathfrak{h}), I(\mathfrak{h}'))$ for $\mathfrak{h}, \mathfrak{h}'$ in $\mathscr{L}$, because $\mathfrak{h} \cap \mathfrak{h}'$ is the largest element of $\mathscr{L}$ contained both in $\mathfrak{h}$ and in $\mathfrak{h}'$.

2) It can be shown that every subspace of finite dimension of $g$ which is stable under the mapping $D \mapsto D^p$ is also stable under the bracket operation.