  Given an $L_{\infty}$-algebra $V$ and an $L_{\infty}$-subalgebra $W$, we give
sufficient conditions for all small Maurer-Cartan elements of $V$ to be
equivalent to Maurer-Cartan elements lying in $W$. As an application, we obtain
a stability criterion for fixed points of a Dirac structure (for instance a
twisted Poisson structure), i.e. points where the corresponding leaf is
zero-dimensional. The criterion guarantees that any nearby Dirac structure also
has a fixed point.
