  We search an identity basis for the adjoint Lie algebra of the algebra
$M_{1,1}(K)$ over a field, where $K$ is either the infinite generated Grassmann
algebra $E$ or $E^1$, the variant of the algebra with $1$. In particular, we
prove that in the case of an infinite base field of characteristic different
from two the identities of $M_{1,1}(E^1)$ are exactly all the consequences of
the identity $[x,y,[z,t],v]=0$. We also find an identity basis of $M_{1,1}(E)$
consisting of three identities.
