Division Algebras, Lie Algebras, Lie Groups: 11

Division Algebras, Lie Algebras, Lie Groups and Spinors

11. Adjoint Octonions, Clifford Algebras and Spinors

Consequently O_L is 64-dimensional, and it is isomorphic to R(8) = CL(0,6). The spinor
space of R(8) = CL(0,6) is the space of 8x1 real column matrices; the spinor space of
O_L = CL(0,6) is O itself. In both cases the spinor space is 8-dimensional.

A 1-vector basis for O_L =CL(0,6) is e_Lp, p=1,...,6 (as we saw above, the 7-vector basis
is the element e_L7). The 2-vector basis of CL(p,q) can be thought of as the Lie algebra
so(p,q), hence

so(0,6) = so(6) --> {e_Lpq, p,q=1,...,6}.

Two other Lie algebras while we're at it.

so(7) --> {e_Lab, a,b=1,...,7};
so(8) --> {e_Lab, e_Lc, a,b,c=1,...,7}.