A note on Lie and Jordan structures of Leavitt path algebras

Let $L_K(E)$ be the Leavitt path algebra of a directed graph $E$ over a field $K$. In this paper, we determine $E$ and $K$ for the Lie algebra ${\mathbf{K}}_{L_K(E)}$ and the Jordan algebra ${\mathbf{S}}_{L_K(E)}$ arising from $L_K(E)$ with respect to the standard involution to be solvable.