If one looks at the multiplication tables for the sets of matrices {I, P, G} and {N, P, G} one sees that the second set has more zeros which suggests that it be better for calculating repeated rotations of R. This is like finding the powers of a complex number with G playing the role of i. The multiplication tables are independent of G and consequently so are the formulas which use the variables α and β, the direction cosines in the plane.
Supplemental (Oct 1): The multiplication table for the set {I, P, G} indicates that the algebraic structure is a ring rather that a group since there is no multiplicative inverse. It acts a lot like numbers of the form a + b√q where q is some integer.
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