  Let $G$ be the classical group, and let Hom$(\mathbb{Z}^m,G)$ denote the
space of commuting $m$-tuples in $G$. Baird proved that the cohomology of
Hom$(\mathbb{Z}^m,G)$ is identified with a certain ring of invariants of the
Weyl group of $G$. In this paper by using the result of Baird we give the
cohomology ring of Hom$(\mathbb{Z}^2,G)$ for simple Lie group $G$ of rank 2.
