At first Evans refers to a representation of the lhs of eq. [1,(1)], of R^a_b Ù q^b, by a cyclic sum using q^b = q^b_ρ dx^ρ and R^a_b q^b_ρ = R^a_ρ:

                R^λ_ρμν + R^λ_νρμ + R^λ_μνρ = cyclic sum of definitions (3)                                 [1,(7)]

taken from

(2)                 R^a_b Ù q^b = R^a_[ρμν] dx^ρÙdx^μÙdx^ν = ½ (R^a_ρμν + R^a_νρμ + R^a_μνρ) dx^ρÙdx^μÙdx^ν

for a = λ.

Then Evans introduces the coefficients of the Hodge dual of the curvature 2-form R^a_b:

The Hodge dual of eq. [1,(3)] is:

                R^{~ λ}_ρ^αβ = ½ ||g||^½ Î^αβμν R^λ_ρμν                                 [1,(5)]

This Evans' original equation can be replaced with the equivalent (but more convenient) equation

(3)                 R^{~ a}_{b αβ} = Î_αβ^μν R^a_{b μν}

which would yield

(4)                 R^{~ a}_{b αβ} dx^αÙdx^β = R^a_{b μν} Î_αβ^μν dx^αÙdx^β = R^a_{b μν} (dx^μÙdx^ν)^~ = (R^a_{b μν} dx^μÙdx^ν)^~.

This means that the 2-form R^{~ a}_{b αβ} dx^αÙdx^β is the Hodge dual of the 2-form R^a_{b μν} dx^μÙdx^ν for each pair of fixed indices a,b.

However, this equation has to be applied to eq. [1,(7)], which seems to be simple due to the short hand notation of that equation. But its correct detailed notation in eq. (2) reveals a problem: In contrast to Evans' belief the transition to

                R^{~ λ}_ρμν + R^{~ λ}_νρμ + R^{~ λ}_μνρ = cyclic sum of definitions (14)                                 [1,(15)]

is not possible by a mere multiplication of eq. [1,(7)] by an Î-factor due to the wrong positioned index ρ at the two right terms in eq. [1,(7)]. In other words: The multiplication of ½ (R^λ_ρμν + R^λ_νρμ + R^λ_μνρ) by

(6)                 (dx^μÙdx^ν)^~ Ù dx^ρ = Î^μν_αβ dx^αÙdx^β Ù dx^ρ

yields

                    ½ (R^a_ρμν + R^a_νρμ + R^a_μνρ) Î^μν_αβ dx^αÙdx^β Ù dx^ρ
(7)                                = ½ R^a_{b μν} Î^μν_αβ dx^αÙdx^β Ùq^b + ½ (R^a_νρμ + R^a_μνρ) Î^μν_αβ dx^αÙdx^β Ù dx^ρ
                                    = ½ R^~a_{b μν} Ùq^b + ½ (R^a_νρμ + R^a_μνρ) Î^μν_αβ dx^αÙdx^β Ù dx^ρ

The first summand yields half of the aspired 3-form R^~a_b Ù q^b whilst the red marked summands cannot be rewritten as duals of R^a_b due to the occurrence of the index ρ at an improper second or third position.

Therewith Evans' proof in [1] has broken down without any chance of repair.

Final Remark
There is an error between Evans' eqs. [1,(19)] and [1,(20)]: The latter is NOT a consequence of the preceeding equation as asserted by Evans:

                D_ρT^~_μ^λ_ν + D_νT^~_ρ^λ_μ + D_μT^~_ν^λ_ρ = − (R^~_μ^λ_ν + R^~_ρ^λ_μ + R^~_ν^λ_ρ)                 [1,(19)]

even if the symbol ~ is removed. That is due to the repeated upper and lower indices μ in eq.[1,(20)]. This repetition does not occur in [1,(19)]:

                D_μT^λμν = − R^λ_μ^μν                                                                                 [1,(20)]

Both sides of this equation are two-tensors (index μ contracted) while eq.(19) displays four-tensors. Since 2 ≠ 4 both equations cannot be equivalent.

This remark is important since in a further note [2, eqs.(25-26) and (39-40)] falsly considers eq. [1,(20)] as the equivalent of the 1st Bianchi identity (the first equ. of (1)).