Let $A$ be an abelian variety over a field and let $L$ be a non-degenerate line bundle on $A$.
Then $L$ gives rise to a morphism $\lambda:A\to A^*$ from $A$ to its dual. As usual, let $K(L):=\ker(\lambda)$. The Cartier dual $K(L)^*$ of $K(L)$ is canonically isomorphic to the kernel of the dual isogeny $\lambda^*:A^{**}=A\to A^*$, which coincides with $\lambda$ (see eg http://van-der-geer.nl/~gerard/AV.pdf, chap. 7). This provides an isomorphism $K(L)\simeq K(L)^*.$
On the other hand, $L$ gives rise to a theta group $G(L)$ with presentation $$ 0\to G_m\to G(L)\to K(L)\to 0. $$ This presentation gives rise to a commutator pairing $e_L:K(L)\times K(L)\to G_m$, which measures the lack of commutativity of $G(L)$. The non-degeneracy of $L$ translates into the non-degeneracy of $e_L$, giving another isomorphism $K(L)\simeq K(L)^*$.
My question is the following. I was under the impression that these two self-dualities of $K(L)$ coincide. I was not able to find a reference for this though. I would be grateful for any pointers. A partial compatibility is proven in Mumford's book on abelian varieties, p. 228 (5).
