Scholl's expository paper "Classical Motives" cites Weil's "Sur les courbes algebriques et les varieties qui s'en deduisent," which I have no access to, for the following result:
If $X$ and $X'$ are smooth projective curves with Jacobian varieties $J,J'$, then $A^1(X\times X') = A^1(X)\bigoplus A^1(X')\bigoplus \operatorname{Hom}(J,J')\bigotimes \mathbb{Q}.$
$A^1(X)$ and $A^1(X')$ arise from the pullbacks of the projection maps, but I have no clue where the $\operatorname{Hom}(J,J')$ comes from. My first guess would be that it comes from some universal property of the Jacobian, but the only one I'm familiar with is that the Jacobian is a coarse moduli space for degree 0 line bundles, which I can't see how to use here.