We quote the essential part of that "proof" starting with

        o q^a_λ = ∂^μ (Γ^ν_μλ q^a_ν − ω^a_μb q^b_λ)                                 (8)

Now define

        R = q^λ_a ∂^μ (Γ^ν_μλ q^a_ν − ω^a_μb q^b_λ)                                 (9)

        (Note that all indices here are dummy (or umbral) indices.)

and use

        q^a_λ q^λ_a = 1                                                                 (10)

This is one of Evans' favorite errors: The left hand expression is the trace of the 4×4 unit matrix and thus has the value 4.

However, regardless if 1 or 4, what follows now is another wrong step that Evans applies in hopeless situations: He wants to resolve Eq.(9) for the expressions

        Q^a_λ := ∂^μ (Γ^ν_μλ q^a_ν − ω^a_μb q^b_λ)

which appear on the right hand side of Eq.(8). So he multiplies Eq.(9) by q^a_λ hoping that so the factor q^λ_a in Eq.(9) would be compensated and removed due to Eq.(10).

However, this operation would require free indices a and λ which both are dummies.