Let {σj} be the three Pauli matrices. {iσj} serve to generate a Clifford algebra C over the reals. Let γj = iσj for j = 1, 2, 3. I emphasize that the algebra is over the reals despite the appearance of i in the expressions. i is not a scalar here; i is not a member of K or even C.

Let me say the same thing very pedantically in order to alleviate some confusion. Let MC2 be the vector space of 2 by 2 complex matrices. Consider the smallest subspace C of MC2 which includes each {σj} for j = 1, 2, 3, and is closed under multiplication, addition and scalar multiplication by reals. C is isomorphic φ to a real rank 3 Clifford algebra where φ(σj) = γj for j = 0, 1, 2, 3.

P = {Σj ajσj | for j = 0, 1, 2, 3; aj is real} where σ0 = I.

There is an associative algebra P generated by {I, σ1, σ2, σ3} which is strategic to quantum mechanics. P is the even part of C. C includes reflections; P does not. Note that V (the span of {iσj}) is not in P; only Hermitian matrices belong to P. You might get away with saying that iσ1 represents a reflection in the x direction but I doubt that this is warranted.

To be a Clifford algebra it suffices that γjγj = −1 and γjγk + γkγj = 0 when j≠k. These are all satisfied. The corresponding Clifford group provides the rigid rotations of R3.

Perhaps it would be clearer to say that the ring of matrices generated by the Pauli matrices is isomorphic to Cl(3) with {iσj} playing the role of Clifford generators γj. Multiplication of matrices goes over to multiplication of Clifford numbers and ditto addition. Some of the confusion is in viewing Pauli matrices as an algebra over the reals, but the complex numbers are themselves are an algebra over the reals, indeed they are Cl(1).

Perhaps an isomorphism table will help;
PauliClifford
0I0
I1
iσjγj
iIγ1γ2γ3(=k)
σj−kγj
σ1γ3γ2
σ2γ1γ3
σ3γ2γ1
In the left column ‘I’ denotes the 2 by 2 identity matrix.

This is a twisted isomorphism! Perhaps there is a better one. The two red entries are not even in the space P.

I subsequently found section II of this. This paper uses the rule γiγj + γjγi = 4δij for the algebra Cl3,0. This is contrary to all the other papers I have seen where γ1γ1 = −1. His rule works for what I would call Cl0,3 except for a factor of two.

I want to do the same thing with the Dirac matrices but I can’t improve on this page which shows that the ‘gamma matrices’ already form a Clifford algebra for a vector space with quadratic form with signature (+ − − −). Even the convention of naming the basis elements γ carries over. Here are further gropings for Dirac.

P.S.

The original Clifford algebras require γjγj = −1 but Minkowski space has a non-definite signature for the metric and describing boosts for that space requires a ‘Clifford algebra’ where γjγj = ±1, depending on j. The definition has thus been stretched so that γjγj = 1 yields a new sort of Clifford algebra. I don’t know whether Clifford considered it. That algebra also yields the orthogonal rotations of R3.