 Two-sided regular representation decomposition

Let $G\times G$ act on $\mathbb{C}G$ by $(g,h)x=gx{h}^{-1}$. Its character is zero unless $g$ and $h$ are conjugate, in which case it equals $|C_G(g)|$. Character column orthogonality gives the same character forwhere the $V_i$ run through the irreducible complex representations. Maschke's theorem therefore gives an isomorphism of the two representations.