By Igor Glaskov
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Extra info for The Budapest for the Tournament Player - 1992 publication.
4) More precisely, C(V) is defined as the quotient of the tensor algebra Q9 V of V (with unit) by the two-sided ideal I, generated by the set of the elements v 0 v + Q (v, v), with v E V. 4) is equivalent to v·w+w·v==-2Q(v,w), v,wEV. 5) If one works over the complex numbers, then the rank of Q determines the Clifford algebras up to isomorphism, whereas in the real case the signature also matters. We will follow the original convention of Clifford  (whose point is that V can have any dimension) and take Q to be positive definite.
J~ - J~ . 1) that r'(a) : ( ~ 2 [Q (~, () J~ - Q (J~, () ~] Chapter 4. The Spin Group and the Spin-c Group 38 belongs to the Lie algebra u(V) of U (V). Running through the various identifications, we see that c (a) == A' (T' (a)) . 5) We also get that T'(a), considered as a complex linear transformation in V, has complex trace equal to 2i Q(~, ~). Therefore a - ~ tracec T'(a) == a ~ i Q (~, ~) == ~ · J~ E spin(V). 7) == iRis the Lie algebra of U(l), the unit circle in C, viewed as a subset of C(V) 0 C.
The Budapest for the Tournament Player - 1992 publication. by Igor Glaskov