Let $A$ be a principally polarized abelian variety over $\mathbf{Q}$. Let $G$ be the Mumford--Tate group of $A$.
The action of complex conjugation on $A(\mathbf{C})$ induces an involution on the de Rham cohomology (with coefficients in $\mathbf{C}$), and thus defines an involution $\iota$ of $G(\mathbf{C})$. Because $A$ is principally polarized, and since complex conjugation preserves the polarization (up to scalar), $\iota$ defines an element of the group $$H:= \mathrm{normalizer}(G(\mathbf{C}) \subseteq \mathrm{GSp}_{2g}(\mathbf{C})).$$ My question, which I will try to make slightly more precise below, is: to what extent is the image of $\iota$ in $H$ independent of $A$ (given $G$)?
The de Rham cohomology of $A$ admits a Hodge decomposition: $$H^1_{dR}(A(\mathbf{C}),\mathbf{C}) = H^{1,0} \oplus H^{0,1}$$ on which complex conjugation sends $H^{1,0}$ to $H^{0,1}$. If $H = \mathrm{GSp}_{2g}$, then this is enough to determine $\iota \in H$ up to conjugacy. Specifically, it is the conjugacy class of involutions whose action on the adjoint representation has the smallest ($=$ most negative) trace.
The only other example I can compute: If $E$ has CM by $K$, then $G = \mathrm{Res}_{K/\mathbf{Q}}(\mathbf{G}_m)$, and so $G(\mathbf{C}) = \mathbf{C}^{\times} \oplus \mathbf{C}^{\times}$. If $H \subset \mathrm{GL}_2(\mathbf{C})$ is the normalizer of $(\mathbf C^{\times})^2$, then $\iota$ also defines a unique conjugacy class of $H$, which can be specified uniquely by saying that it generates the Weyl group.
More generally, one is tempted to ask whether $\iota$ defines a specific conjugacy class of involutions in $H$, except that perhaps one should ask for the image of $\iota$ in the slightly larger class of involutions in $H$ modulo conjugation by $G(\mathbf{C})$. Or perhaps there is another formulation, any suggestions welcome.
