On balanced bases

It is proved that either a given balanced basis of the algebra $(n+1)M_1\oplus M_n$ or the corresponding complementary basis is of rank $n+1$. This result enables us to claim that the algebra $(n+1)M_1\oplus M_n$ is balanced if and only if the matrix algebra $M_n$ admits a WP-decomposition, i.e., a family of $n+1$ subalgebras conjugate to the diagonal algebra and such that any two algebras in this family intersect orthogonally (with respect to the form $\operatorname{tr}XY$) and their intersection is the trivial subalgebra. Thus, the problem of whether or not the algebra $(n+1)M_1\oplus M_n$ is balanced is equivalent to the well-known Winnie-the-Pooh problem on the existence of an orthogonal decomposition of a simple Lie algebra of type $A_{n-1}$ into the sum of Cartan subalgebras.