  In this paper, we show that a unital simple Steinberg algebra is central, and
a nonunital simple Steinberg algebra has zero center. We identify the fields
$K$ and Hausdorff ample groupoids $\mathcal{G}$ for which the simple Steinberg
algebra $A_K(\mathcal{G})$ yields a simple Lie algebra $[A_K(\mathcal{G}),
A_K(\mathcal{G})]$. We apply the obtained results on simple Leavitt path
algebras, simple Kumjian-Pask algebras and simple Exel-Pardo algebras to
determine their associated Lie algebras are simple. In particular, we give
easily computable criteria to determine which Lie algebras of the form
$[L_K(E), L_K(E)]$ are simple, when $E$ is an arbitrary graph and the Leavitt
path algebra $L_K(E)$ is simple. Also, we obtain that unital simple Exel-Pardo
algebras are central, and nonunital simple Exel-Pardo algebras have zero
center.
