Let $\Delta$ be a quaternion algebra over $\mathbf Q$ and let $\mathcal O_\Delta$ be a maximal order in $\Delta$. Recall that a false elliptic curve over a scheme field $S$ K$ is a pair $(A/S,i)$ (A/K,i)$ consisting of an abelian surface $A/S$ A/K$ and a ring homomorphism $i : \mathcal O_\Delta\to End_S(A)$End_K(A)$. Suppose that $\Delta$ is indefinite, i.e., $\Delta\otimes\mathbf R \simeq M_2(\mathbf R)$. There is an involution $*:\Delta\to\Delta$ that coincides with taking transpose under the previous isomorphism. It is well-known (easy?) that if $S$ K$ is a $\mathbf Q$-scheme of characteristic zero there is a polarization of $A/S$ A/K$ such that the corresponding Rosatti involution in $End_S(A)\otimes\mathbf End_K(A)\otimes\mathbf Q$ corresponds to $* : \Delta\to\Delta$ by $i$. Moreover, this polarization is unique up to a rational number.
Question 1: is it possible to find a (necessarily unique) principal polarization with this property?
Question 2: is it possible to find such a principal (principal) polarization over a general scheme field $S$?K$ of non-zero characteristic?