  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.
