Characterization of linear and alinear quasigroups

A quasigroup $(Q,\,\cdot\,)$ is said to be linear [alinear] if, for all $x,y\in Q$, $xy=\phi x+c+\psi y$, where $(Q,+)$ is some group, $\phi$ and $\psi$ are its automorphisms[antiautomorphisms], $c\in Q$. We prove that (primitive) linear [alinear] quasigroups are characterized by one identity in four variables.