  For a field $F$ of characteristic not 2 and a directed row-finite graph
$\Gamma$ let $L(\Gamma)$ be the Leavitt path algebra with the standard
involution $*.$ We study the Lie algebra $K=K(L(\Gamma),*)$ of
$*-$skew-symmetric elements and find necessary and sufficient conditions for
the Lie algebra $[K,K]$ to be simple.
