By Eisenhart L. P.

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M = P (Q(F )\{0}), image into P (F ) of the isotropic cone minus its origin of F , is m-dimensional and called the compactiﬁcation of V = Er,s . M is identical to the homogeneous space PO(F )/Sim(V ),41 quotient group of P O(F ) = O(r + 1, s + 1)/Z2 by the group SimV of similarities of V . 40 Directly for C , there are four basis elements 1, e , e , and e e with e2 = 1, e2 = 11 1 2 1 2 1 2 −1, e1 e2 = −e2 e1 , (e1 e2 )2 = 1, (e1 e2 )e1 = −e2 , (e1 e2 )e2 = −e1 . If we map 1 → 10 01 0 −1 1 0 , e1 → , e2 → , and then e1 e2 → , we get an algebra 01 10 1 0 0 −1 isomorphism between C11 and m(2, R).

Let {e1 , . . , er , er+1 , . . , er+s } be the standard orthonormal basis of V and {e0 , en+1 } be a basis of H such that for any x in H, x = x 0 e0 + x n+1 en+1 , (x|x) = (x 0 )2 − (x n+1 )2 . The equation of the cone Q(F ) is the following n+1 r x = (x 0 , x 1 , . . , x n+1 ) ∈ Q(F ) if and only if i=0 The Euclidean sphere of radius the following equation: (x l )2 = 0. (x i )2 − l=r+1 √ 2 associated with the basis {e0 , . . , en+1 } of F has r n+1 (x i )2 + i=0 (x l )2 = 2. l=r+1 x belongs to the intersection of Q(F ) and of the sphere if and only if r n+1 (x i )2 = i=0 (x j )2 = 1, j =r+1 that is, if and only if x belongs to the product of the unit sphere S r of the standard Euclidean space Er+1 , with the basis {e0 , .

By definition, (q) = (−1) n(n−1) 2 n ai (mod(K × )2 ) i=1 is the “discriminant” of q. The construction of a Clifford algebra associated with a quadratic regular space (E, q) is based on the fundamental idea of taking the square root of a quadratic form, more precisely of writing q(x) as the square of a linear form ϕ on E such that for any x ∈ E, q(x) = (ϕ(x))2 . 2 Clifford Mappings Let A be any associative algebra with a unit element 1A . 1 Definition A Clifford mapping f from (E, q) into A is a linear mapping f such that for any x ∈ E, (f (x))2 = q(x)1A .