  We recall that a smooth ample surface $\mathcal{S}$ in a general
$(1,2,2)$-polarized abelian threefold, which is the pullback of the Theta
divisor of a smooth plane quartic curve $\mathcal{D}$, is a surface isogenous
to the product $\mathcal{C} \times \mathcal{C}$, where $\mathcal{C}$ is a genus
$9$ curve embedded in $\mathbb{P}^3$ as complete intersection of a smooth
quadric and a smooth quartic. We show that the space of global holomorhic
sections of the canonical bundle of this surface is generated by certain
determinantal bihomogeneous polynomials of bidegree $(2,2)$ on $\mathbb{P}^3$,
which can be used to define biquadratic addition laws on the Jacobi model of
elliptic curves, embedded in $\mathbb{P}^3$ as complete intersection of two
quadrics. Finally, we use this interesting relationship with the biquadratic
addition laws to describe the behavior of the canonical map of $\mathcal{S}$.
