Nonlinear realisations of Lie superalgebras

For any decomposition of a Lie superalgebra $\mathcal{G}$ into a direct sum $\mathcal{G}=\mathcal{H}\oplus \mathcal{E}$ of a subalgebra $\mathcal{H}$ and a subspace $\mathcal{E}$, without any further resctrictions on $\mathcal{H}$ and $\mathcal{E}$, we construct a nonlinear realisation of $\mathcal{G}$ on $\mathcal E$. The result generalises a theorem by Kantor from Lie algebras to Lie superalgebras. When $\mathcal{G}$ is a differential graded Lie algebra, we show that it gives a construction of an associated $L_{\mathrm{\infty }}$-algebra.