 If a matrix is an idempotent matrix, it must lie in such a P"m" and in this case P"m" is isomorphic to the ring of split-complex numbers.The case of a nilpotent matrix, "mm" = 0, arises when only one of "b" or "c" is non-zero, and the commutative subring P"m" is then a copy of the dual number plane.When M(2, R) is reconfigured with a change of basis, this profile changes to the profile of split-quaternions where the sets of square roots of and − take a symmetrical shape as hyperboloids.