Let $G$ be a connected reductive group over a (perfect, why not) field $F$. Let $m$, $pr_1$, $pr_2$ denote the multiplication, first, and second projection maps from $G \times G$ to $G$.

Then I'm pretty sure that I can prove the following fact: if $L$ is a line bundle on $G$, then $m^\ast(L)$ is isomorphic to $pr_1^\ast(L) \cdot pr_2^\ast(L)$. This (plus Hilbert's 90) implies that $Pic(G)$ classifies the central extensions of $G$ by the multiplicative group $G_m$, by some stuff in SGA 7, I believe.

The way that I can prove the above fact is by using Kottwitz's isomorphism, which describes $Pic(G)$ in terms of the dual group. I'll probably include this Kottwitzish proof in something I'm writing, but I'm left with the following question:

Is there a proof in the literature that $m^\ast(L)$ is isomorphic to $pr_1^\ast(L) \cdot pr_2^\ast(L)$ for line bundles over reductive groups? Someone must have written this up 30 years ago, right? And the implication that $Pic(G)$ classifies central extensions by $G_m$? Is this published somewhere? It certainly shouldn't require passage to the dual group!

Of course, if I've messed something up, and the above fact is false, I'd appreciate such information too!