Complemented subspaces of Banach spaces C(K× L)

We prove that, for every compact spaces K₁,K₂ and a compact group G, if both K₁ and K₂ map continuously onto G, then the Banach space C(K₁× K₂) contains a complemented subspace isometric to $C(G)$. Consequently, answering a question of Alspach and Galego, we get that C(βω×βω) contains a complemented isomorphic copy of C([0,1]^κ) for every cardinal number 1≤κ≤ c and hence a complemented copy of $C(K)$ for every metric compact space K.