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

Cohomology of the spaces of commuting elements in Lie groups of rank two