Here I compute the structure coefficients of the Lie algebra for the Clifford group, considering that that group is a Lie group. Well this is the connected component that includes the identity. Actually is it that part of the Clifford group generated by products of unit vectors. I argue here that there are just two Clifford numbers that produce any particular action on V.

The code below inspired me to do this by hand once I knew the answers. The generators of the Clifford group are Gij = C1 + δ γiγj where δ is a small real and i ≠ j. Note that there are no reflections near the identity.

The commutator [Gij; Gkl] = (C1 + δ γiγj)(C1 + δ γkγl) − (C1 + δ γkγl)(C1 + δ γiγj) = 2δ2 (γiγjγkγl − γkγlγiγj) considering that C1 commutes with everything. It is tedious to elaborate all of the combinations but observe the following:
[G01; G23] = 0 (three permutations)
[G01; G02] = −2G12 (12 permutations with various signs).
[G01; G01] = 0 (6 permutations)

The Clifford group, at least that part generated products of 2 unit vectors, is a manifold with an obvious differential metric: how much is a random vector moved by a small rotation? This is a definite integral over S(n−1) which I may evaluate. It is a constant 

(define (g x) (C+ C1 (sm .000001 x)))

(map mag (ng (g (C* (C* g0 g1) (C* g2 g3))))) ; => 4 x 4e-12
(map mag (ng (C* (g (C* g0 g1)) (g (C* g2 g3))))); => 4 x 4e-44
(map mag (ng (C* (g (C* g0 g1)) (g (C* g1 g2))))); => 4 x 4e-44
(define (cm x y) (let ((X (g x))(Y (g y))) (C+ (C* X Y) (C- (C* Y X)))))
(cm (C* g0 g1) (C* g2 g3))

(let ((g01 (C* g0 g1)) (g02 (C* g0 g2)) (g03 (C* g0 g3)) (g12 (C* g1 g2)) (g13 (C* g1 g3)) (g23 (C* g2 g3))) (list
    (cm g01 g02) (cm g01 g03) (cm g01 g12) (cm g01 g13) (cm g01 g23)
    (cm g02 g03) (cm g02 g12) (cm g02 g13) (cm g02 g23)
    (cm g03 g12) (cm g03 g13) (cm g03 g23)
    (cm g12 g13) (cm g12 g23)
    (cm g13 g23))) ; =>
(((((0 . 0) 0 . 0) (0 . 0) -2e-12 . 0) ((0 . 0) 0.0 . 0) (0.0 . 0) 0 . 0) -x6
((((0 . 0) 0 . 0) (0 . -2e-12) 0 . 0) ((0 . 0.0) 0 . 0) (0.0 . 0) 0 . 0) -x5
((((0 . 0) 0 . 0) (0 . 0) 0.0 . 0) ((0 . 0) 2e-12 . 0) (0.0 . 0) 0 . 0) xa
((((0 . 0) 0 . 0) (0 . 0.0) 0 . 0) ((0 . 2e-12) 0 . 0) (0.0 . 0) 0 . 0) x9
((((0 . 0) 0 . 0.0) (0 . 0) 0 . 0) ((0 . 0) 0 . 0) (0.0 . 0) 0 . 0.0) 0
((((0 . 0) 0 . -2e-12) (0 . 0) 0 . 0) ((0 . 0.0) 0.0 . 0) (0 . 0) 0 . 0) -x3
((((0 . 0) 0 . 0) (0 . 0) 0.0 . 0) ((0 . 0) 0.0 . 0) (-2e-12 . 0) 0 . 0) -xc
((((0 . 0) 0 . 0) (0 . 0.0) 0 . 0) ((0 . 0) 0.0 . 0) (0 . 0) 0 . 0.0) 0
((((0 . 0) 0 . 0.0) (0 . 0) 0 . 0) ((0 . 2e-12) 0.0 . 0) (0 . 0) 0 . 0) x9
((((0 . 0) 0 . 0) (0 . 0) 0.0 . 0) ((0 . 0.0) 0 . 0) (0 . 0) 0 . 0.0) 0
((((0 . 0) 0 . 0) (0 . 0.0) 0 . 0) ((0 . 0.0) 0 . 0) (-2e-12 . 0) 0 . 0) -xc
((((0 . 0) 0 . 0.0) (0 . 0) 0 . 0) ((0 . 0.0) -2e-12 . 0) (0 . 0) 0 . 0) -xa
((((0 . 0) 0 . -2e-12) (0 . 0.0) 0.0 . 0) ((0 . 0) 0 . 0) (0 . 0) 0 . 0) -x3
((((0 . 0) 0 . 0.0) (0 . 2e-12) 0.0 . 0) ((0 . 0) 0 . 0) (0 . 0) 0 . 0) x5
((((0 . 0) 0 . 0.0) (0 . 0.0) -2e-12 . 0) ((0 . 0) 0 . 0) (0 . 0) 0 . 0)) -x6
(define (ng x) (map (lambda (b) (let ((y (turn b x))) (C+ (C- y) (gs y)))) basis))