  We establish a connection between the problem of constructing maximal
collections of mutually unbiased bases (MUBs) and an open problem in the theory
of Lie algebras. More precisely, we show that a collection of m MUBs in K^n
gives rise to a collection of m Cartan subalgebras of the special linear Lie
algebra sl_n(K) that are pairwise orthogonal with respect to the Killing form,
where K=R or K=C. In particular, a complete collection of MUBs in C^n gives
rise to a so-called orthogonal decomposition (OD) of sl_n(C). The converse
holds if the Cartan subalgebras in the OD are also *-closed, i.e., closed under
the adjoint operation. In this case, the Cartan subalgebras have unitary bases,
and the above correspondence becomes equivalent to a result relating
collections of MUBs to collections of maximal commuting classes of unitary
error bases, i.e., orthogonal unitary matrices.
  It is a longstanding conjecture that ODs of sl_n(C) can only exist if n is a
prime power. This corroborates further the general belief that a complete
collection of MUBs can only exist in prime power dimensions. The connection to
ODs of sl_n(C) potentially allows the application of known results on (partial)
ODs of sl_n(C) to MUBs.
