For any field $K$ and integer $n\geq 2$ we consider the Leavitt algebra $L =
L_K(n)$. $L$ is an associative algebra, but we view $L$ as a Lie algebra using
the bracket $[a,b]=ab-ba$ for $a,b \in L$. We denote this Lie algebra as $L^-$,
and consider its Lie subalgebra $[L^-,L^-]$. In our main result, we show that
$[L^-,L^-]$ is a simple Lie algebra if and only if char$(K)$ divides $n-1$. For
any positive integer $d$ we let $S = M_d(L_K(n))$ be the $d\times d$ matrix
algebra over $L_K(n)$. We give sufficient conditions for the simplicity and
non-simplicity of the Lie algebra $[S^-,S^-]$.