What's written below is a proof and discussion of the following:
Let ga, a=1,...,7, be seven anticommuting 8x8
real matrices satisfying
ga2 = -I,
where I is the 8x8 identity matrix. Let X be a nonzero 8x1 real column matrix.
Then for all distinct a,b = 1,...,7, there exists a unique AXab linear in
the ga such that
gbX = AXabX.
This is an octonion product.
A related result, which will not be proved, is the following:
Let ga, a=1,...,7, and X be as above, and let U be
a linear in the ga
and I (U is assumed nonzero and is necessarily invertible). Then for all distinct
a,b = 1,...,7, there exists a unique BUXab linear in the ga and I
gbX = BUXabX.
gb = BUXab.
This is an octonion product with U-1 the product identity (this is
a matrix way of creating an X,Y-product).
SO(8), Spinors, and Octonions.
For A and X arbitrary octonions define the O --> O maps
AL[X] = AX and AR[X] = XA.
eLab...c[X] = ea(eb(...(ecX)...)),
eRab...c[X] = (...((Xea)eb)...)ec.
It is not hard to prove that the span of the set of all such maps is isomorphic
to the associative algebra R(8) (associativity is built in - see the
In particular, the distinct elements of the form
eL0, dim = 1,
eLa, dim = 7,
eLab, dim = 21,
eLabc, dim = 35,
where a,b,c = 1,...,7, form a basis for the algebra of left adjoint actions
of O on O, denoted OL.
Well, OL ~ R(8) (isomorphism), so OL is a representation
of the Clifford algebra Cl0,6 of the real 6-dimensional orthogonal space
with negative definite metric. We can make the following multivector basis assignments:
eLp, (1-vectors), p = 1,...,6;
eLpq, (2-vectors), p,q = 1,...,6;
eLpqr, (3-vectors), p,q,r = 1,...,6;
eLpq7, (4-vectors), p,q = 1,...,6;
eLp7, (5-vectors), p = 1,...,6;
The spinor space of CL0,6 ~ R(8) consists of the space of columns of
8 real numbers. The spinor space of CL0,6 ~ OL is the 8-dimensional
space O on which OL acts.
NOTE: most of the errors people make in thinking about the octonions and their
relationship to Lie groups like SL(2,O) come from trying to do Lie group things with O
instead of OL; O is a spinor space in this context, and its
nonassociativity makes it an inappropriate choice for trying to generate associative algebras;
this is the message of the mathematics; OL (OR = OL
(equality - not just isomorphism)) is an inherent part of the mathematics of O.
Now that we've established the roles everything is to play, consider the subspace of OL
spanned by the elements eLa and eLab, a,b = 1,...,7. Given the commutator product
this subspace is isomorphic to so(8), the Lie algebra of the orthogonal group SO(8).
The subspace spanned by the elements eLab, a,b = 1,...,7
(it is assumed a and b are distinct), generates an so(7) Lie subalgebra of so(8).
Let X be an arbitrary nonzero element of O, an OL spinor), and let X-1 be its inverse
(if X is a unit octonion then this is just the conjugate). Note:
eLab[X] = ea(ebX) = (ea(ebX))X-1X
(eaX-1)(Xeb) is the X-product of ea and eb and is linear in ec,
c = 1,...,7; and it is unique, given X, ea and eb. This implies that for all elements A of so(7)
(linear in the eLab), and for all nonzero octonions X, there exists a unique so(8) element AX
linear in the eLa, a = 1,...,7, such that
A[X] = AX[X].
This gives us a way of deriving multiplication tables for O starting from a matrix representation of
Cl0,6. Let gp, p=1,...,6, be an anticommuting basis
for the space of 1-vectors (8x8 real matrices), and define
which also anticommutes with the gp. For distinct a,b = 1,...,7, define
The set of distinct ga and gbc, a,b,c = 1,...,7,
form a basis for the Lie algebra so(8).
Choose a nonzero spinor X (a column of 8 real numbers). For all distinct a,b = 1,...,7, there exists
a unique AX,a,b = a1 g1 + ... +
a7 g7 such that
gabX = AX,a,bX.
For all distinct a,b define the product
It should now be somewhat clear that the set of ga, a = 1,...,7, together with
the ox-product, form a basis for the hypercomplex part of a copy of the octonions
(extend the basis to the identity, and the product in the obvious ways, to complete the picture).
Analogous but simplified reasoning on Cl0,1 and Cl0,2 yield the complex and quaternion algebras.
I don't believe it would be difficult to prove that this trick would not work on Cl0,r, with r =/= 1,2,6.
Octoshop III page.