I share a view with many others that the connection of mathematics to physics is not accidental, and I share an intuition with perhaps fewer that existence is attributable in part to mathematical resonance - that the rich and specific structure of our unique physical reality mirrors the rich and specific structure of the few mathematical resonances. What follows is mathematics, as pure as I can make it given my knowledge and energy. What I just wrote was philosophy, devoid of rigor, and easily rejected. Rejection will not effect the mathematics, only its interpretation. However, that definition assumes the underlying algebra is associative and alternative.
T is neither.
By the way, an idempotent is primitive if not the sum of two other nonzero idempotents. That is, it has been proven only in part that the general form of this resolution is unique. Suppose that G m is the result of transforming D m . K D m , K an element of T . For the sake of brevity, and because I spent 2 years organizing these thoughts into a somewhat expensive book , I'll quit here.