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 $L_\infty$-algebra.