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 $\mu_t=\mu_0+t\varphi_1+t^2\varphi_2$ where $\mu_t$ is a Lie algebra on $V \otimes K[[t]]$. It is equivalent to say that we have the following system $\sum_{i+j \leq 4} \varphi_i \circ \varphi_j= 0$. 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 $\mathcal{H}_{2p+1}$.