The special case $x=y$ is just the notion of a so-called sandwich element (that is, an element $x$ of $L$ such that $(\operatorname{ad}x)^2=0$). Such elements played an important role in the classification of finite-dimensional simple Lie algebras over fields of positive characteristic. For instance, over algebraically closed fields of characteristic $p>5$, in the paper
[A. A. Premet: Lie algebras without strong degeneration, Mat. Sb. (N.S.) 129(171(1))(1986), 140–153]
it is proved that the presence of sandwich elements characterizes finite-dimensional simple Lie algebras which are not classical.
