23/03/13 21:12:59.40 UeELXD7y.net
>>358
つづき
The finish
For a, b in V define B(a, b) = (?ab ? ba)/2. Because of the identity (a + b)2 ? a2 ? b2 = ab + ba, it follows that B(a, b) is real. Furthermore, since a2 <= 0, we have: B(a, a) > 0 for a ≠ 0. Thus B is a positive definite symmetric bilinear form, in other words, an inner product on V.
Let W be a subspace of V that generates D as an algebra and which is minimal with respect to this property. Let e1, ..., en be an orthonormal basis of W with respect to B. Then orthonormality implies that:
e_{i}^{2}=-1,\quad e_{i}e_{j}=-e_{j}e_{i}.
If n = 0, then D is isomorphic to R.
If n = 1, then D is generated by 1 and e1 subject to the relation e2
1 = ?1. Hence it is isomorphic to C.
If n = 2, it has been shown above that D is generated by 1, e1, e2 subject to the relations
e_{1}^{2}=e_{2}^{2}=-1,\quad e_{1}e_{2}=-e_{2}e_{1},\quad (e_{1}e_{2})(e_{1}e_{2})=-1.
These are precisely the relations for H.
つづく