Finite Symmetries, Orientations of Bases, Extended CPT for Gravity and the Standard Model

The author suggested three 8-dimensional vector spaces withdifferent multiplications and number of generators for SI with theSU(3) 8 gluon GellMann matrices λk, the signed quaternion SU(2)generators (Pauli matrices σj, j = 1,2,3 and id) of the weak interactionWI and newly added is in the MINT-Wigris project the octonianswhich have seven Gleason measuring frames GF, not only theweak Pauli spin GF. Developing a unified theory for the four basiinteractions the doubling of quaternion numbers to octonians isnecessary. The use of the GF real cross product was necessary todouble up the complex numbers to quaternions. As projections theλk matrices are 3-dimensional blown up σj matrices with a rowand column of coordinates 0 added. The blown up σ3 matrices arelinearly independent and give only 2 not 3 gluons as geometricalinvariants. The 3 WI invariants are the weak bosons W+, W- andZ° and one more invariant photon for EMI for the electromagneticinteraction with the U(1) symmetry, a circle.