  The paper is devoted to representation theoretic and algebraic geometric
aspects of the theory of orthogonal decompositions of Lie algebra sl(n) into
Cartan subalgebras orthogonal with respect to Killing form and the relevant
theory of mutually unbiased bases. We describe the main steps of our proof for
the existence of 4-dimensional family of orthogonal pairs of Cartan subalgebras
in sl(6). As an application, we prove existence of a real 4-dimensional family
of pairs of mutually unbiased bases in C^6, thus solving the long standing
problem in Quantum Information Theory.
