  We extend Jordan's notion of principal angles to work for two subspaces of
quaternionic space, and so have a method to analyze two orthogonal projections
in M_n(A) for A the real, complex or quaternionic field (or skew field). From
this we derive an algorithm to turn almost commuting projections into commuting
projections that minimizes the sum of the displacements of the two projections.
We quickly prove what we need using the universal real C*-algebra generated by
two projections.

Operator Algebras