Contact and 2-compatible Lie algebras

A $n$-dimensional Lie algebra $g=(V,\mu )$ is called $2$-compatible if it is isomorphic to a quadratic deformation of a Lie algebra $g_0=(V,{\mu }_0)$. By quadratic deformation we means a formal deformation ${\mu }_t={\mu }_0+t{\phi }_1+t^2{\phi }_2$ where ${\mu }_t$ is a Lie algebra on $V\otimes K[[t]]$. It is equivalent to say that we have the following system ${\sum }_{i+j\le 4}{\phi }_i\circ {\phi }_j=0$. This notion naturally appears in the theory of classification of contact Lie algebras because any $(2p+1)$-dimensional contact Lie algebra is isomorphic to a quadratic deformation of the Heisenberg algebra ${\mathcal{H}}_{2p+1}$.