  In this paper, we first study properties of the lower central chains for
Novikov algebras. Then we show that for every Lie nilpotent Novikov
algebra~$\mathcal{N}$, the ideal of~$\mathcal{N}$ generated by the set~$\{ab -
ba\mid a, b\in \mathcal{N}\}$ is nilpotent. We secondly provide necessary and
sufficient conditions on the graph $E$ and the field $K$ for which the Leavitt
path algebra $L_K(E)$ is Lie solvable. Consequently, we obtain a complete
description of Lie nilpotent Leavitt path algebras, and show that the Lie
solvability of~$L_K(E)$ and the Lie nilpotency of $[L_K(E),L_K(E)]$ are the
same.
