Fundamental group of Lie groups

The Eckmann-Hilton argument is the correct answer, but it might be amusing to note that there is a very explicit homotopy as well. Suppose $alpha_1$, $alpha_2 in pi_1 G$, and define

$alpha : I^2 rightarrow G$

by $alpha(t_1,t_2) = alpha_1(t_1) cdot alpha_2(t_2)$, where $cdot$ is the product in $G$. Then along the diagonal, we have $alpha_1 cdot alpha_2$, the product using the group operation, while along the bottom edge followed by the right edge we have the composition $alpha_1 * alpha_2$, the product of loops in the fundamental group. Deforming the path shows they're homotopic. Similarly, along the left edge, followed by the top edge we get $alpha_2 * alpha_1$, so this product is commutative.