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All such structures of Clifford algebras for C + (V ) corresponding to different choices of u define the same conjugation, which is identical to the restriction of τ to C + (V ). 1, which gives explicitly the nature of + , according to r − s modulo 8. Such a result is due to the nature of Cr,s and Cr,s the Brauer–Wall group:35 BW (R) = Z/8Z. We agree to denote by m(n, F ) the real algebra of square matrices of degree n with coefficients in the field F = R, C, or H (the usual noncommutative field of real quaternions).
3 Proposition Let us assume that K = R, C. The Lie algebra spin(E, q) of Spin(E, q) is the Lie subalgebra of the Lie algebra associated with the associative algebra C(E, q)27 consisting of the space C2 (E, q) deﬁned above. spin(E, q) operates 26 Following Deheuvels (R. Deheuvels, Formes Quadratiques et Groupes Classiques, op. ), we denote by RO(q), for a quadratic regular complex or Euclidean real space, the twofold covering group of O(q) (according to the exact sequence 1 → Z2 → RO(q) → O(q) → 1); RO(r, s) the twofold covering group of the standard pseudo-Euclidean real space Er,s with (r, s) = (1, 1) (according to the exact sequence 1 → Z2 → RO(r, s) → O(r, s) → 1).
2 Clifford Algebras 13 H = A(−1, −1), often denoted by ( −1,−1 R ), is the unique division quaternion algebra over the real field R. When K is a local field or an algebraic number field of finite degree, the previous pairing is surjective and complete. Therefore, a central division algebra B over such a field K with an involution J of the first kind is necessarily a quaternion algebra, and any involution of B of the first kind with sign η can be written as q → f −1 q J f , where f belongs to the multiplicative group of units of B and f J = −ηf .
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