I found in Encyclopedia of math
https://encyclopediaofmath.org/wiki/Abelian_surface
there is a claim that:
"A principally polarized Abelian surface $(A,λ)$ is either the Jacobi variety $J(H)$ of a smooth projective curve $H$ of genus $2$, and $λ=θ$ is the class of the theta divisor, or $A$ is the product of two elliptic curves with $λ$ the product polarization."
It seems to be a well-known result, I wonder if there is any book/note/paper with more details on that claim?