Krdytkyu

Let $E/F$ be a CM extension of number fields (i.e., $F$ is totally real and $E/F$ is a totally imaginary quadratic extension). I assume $E=F[alpha]$ with $alphain Ebackslash F$ such that $alpha^2in F$. I denote by $xmapstobar{x}$ the non-trivial element of $Gal(E/F)$.

If $S$ is an $E$-algebra, I have an isomorphism
$$phi: Eotimes_F Ssimeq Stimes S $$
given by mapping $xotimes s$ to $(bar{x}s,xs)$. Its inverse $phi^{-1}$ maps $(s,t)in Stimes S$ to
$alphaotimes frac{t-s}{2alpha} + 1otimes frac{s+t}{2} in Eotimes_F S $ (this doesn't depend on the choice of $alpha$).

Now let $V$ be an $E$-vector space, say of dimension $n=3$ (which is my case of interest, but the answer shouldn't depend on $n$) and denote by $V_S:=Votimes_E S$. My question is:

Is there a canonical way to define an isomorphism (of $S$-modules) between $Votimes_F S$
and $V_Soplus V_S$ ?

My problem is the following: if I choose an $E$-basis $mathcal{B}={e_1,e_2,e_3}$ of $V$ (which is still an $S$-basis of $V_S$ and an $Eotimes_F S$-basis of $Votimes_F S$), then one can define an isomorphism as above by mapping a column vector $X=begin{pmatrix}x_1\ x_2\ x_3end{pmatrix}in (Eotimes_F S)^3simeq_{mathcal{B}} Votimes_F S$, to
$phi(X)=begin{pmatrix}phi(x_1)\ phi(x_2)\ phi(x_3)end{pmatrix}=:(phi(X)_1,phi(X)_2)in S^3times S^3simeq_{mathcal{B}}V_Soplus V_S$.

But if $mathcal{B}'$ is another $E$-basis of $V$, denote by $Pin GL_3(E)$ the transition matrix (whose columns are the coordinates of the vectors of $mathcal{B}$ in $mathcal{B}'$): then a column vector $Xin (Eotimes_F S)^3$ in $mathcal{B}$ is changed into $PX$ in $mathcal{B}'$, thus maps to
$phi(PX)=(phi(PX)_1,phi(PX)_2)=(bar{P}phi(X)_1,Pphi(X)_2)in S^3times S^3$, whereas
$(phi(X)_1,phi(X)_2)$ is changed into $(Pphi(X)_1,Pphi(X)_2)neq (bar{P}phi(X)_1,Pphi(X)_2) $

Is there any way to do things canonically ?

In the end, I am interested in knowing whether there exists a canonical isomorphism between these two modules (which would give a canonical isomorphism between groups $GL(Votimes_F S)$ and $GL(V_S)times GL(V_S)$) such that, if $langle,,,rangle$ is a given $E/F$-hermitian product on $V$, which we can extend to $Votimes_F S$ with values in $Eotimes_S F$, then
for any $gin GL(Votimes_F S)$, for all $v$, $win Votimes_F S$, the quantity
$langle gcdot{}v,gcdot{}wranglein Eotimes_F S$ admits a nice description (i.e., corresponds by $phi:Eotimes_F Srightarrow Stimes S$) as a pair of $S$-bilinear products on $V_S$ whose definition is canonical. I managed to do something, but this implies choosing a basis of $V$ to define $Votimes_F Ssimeq V_Soplus V_S$ and to define the bilinear products on $V_S$ (but the two combined do not depend on $mathcal{B}$, luckily !).
In the very end, I want to show that the subgroup of $GL(Votimes_F S)$ that preserves $langle,,,rangle_{Votimes_F S}$ (which we call $U(V)(S)$ )maps canonically to $GL(V_S)$.

Thanks a lot for your help !