Twistor triangles in the period domain of complex tori

We study the geometry of the (generalized) twistor triangles $\triangle J_1J_2J_3$ in the period domain of compact complex tori of complex dimension $2n$ by the means of the representation theory of the algebras (of real dimension 8) generated by the complex structures $J_1,J_2,J_3$. Considering the period domain as the homogeneous space for $G=GL_{4n}(\mathbb{R})$, we introduce on it a $G$-invariant pseudometric and define pseudometric invariants, helping us to distinguish triangles from a reasonable class up to $G$-equivalence.