  We construct L$_{\infty}$ algebras for general `initial data' given by a
vector space equipped with an antisymmetric bracket not necessarily satisfying
the Jacobi identity. We prove that any such bracket can be extended to a 2-term
L$_{\infty}$ algebra on a graded vector space of twice the dimension, with the
3-bracket being related to the Jacobiator. While these L$_{\infty}$ algebras
always exist, they generally do not realize a non-trivial symmetry in a field
theory. In order to define L$_{\infty}$ algebras with genuine field theory
realizations, we prove the significantly more general theorem that if the
Jacobiator takes values in the image of any linear map that defines an ideal
there is a 3-term L$_{\infty}$ algebra with a generally non-trivial 4-bracket.
We discuss special cases such as the commutator algebra of octonions, its
contraction to the `R-flux algebra', and the Courant algebroid.
