An alternate definition of the Clifford group uses the expression cvα(c)^{−1} in place of cvc^{−1}.
This defines the same set of Clifford numbers but the orthogonal matrices they lead to have the opposite sign when c is odd.

To find those Clifford numbers c that commute with each member of V we first seek all c such that cγ_{0} = γ_{0}c.

Split c thus c = f + γ_{0}g where neither f nor g mention γ_{0}.

cγ_{0} = γ_{0}c ↔
(f + γ_{0}g)γ_{0} = γ_{0}(f + γ_{0}g) ↔
fγ_{0} + γ_{0}gγ_{0} = γ_{0}f + γ_{0}γ_{0}g ↔

γ_{0}f = fγ_{0} & γ_{0}gγ_{0} = γ_{0}γ_{0}g ↔ f and g are both pure even.
(Note that γ_{0} has an inverse: −γ_{0}.)

The code corroborates this.

Said another way, a Clifford number commutes with γ_{0} iff each of its terms mentions an even number of other base vectors.
When we require that a Clifford number commute with each of the base elements we are clearly left with only the reals among the Clifford numbers.