Quotations in black from the Evans paper [1] (also contained in [2]):

. . . The Einstein field equation

        R_μν − ½ R g_μν = kT_μν                                                                 (11)

. . . can be deduced [1-10] as a special case of the Evans field equaton.

        R_μν = R_μ^a q_ν^b η_ab                                                                 (13)

        T_μν = T_μ^a q_ν^b η_ab                                                                 (14)

        g_μν = q_μ^a q_ν^b η_ab                                                                 (15)

together with the reversed equations

        R_μ^a = R_μν q_b^ν η^ab                                                                 (13')

        G_μν = G_μ^a q_ν^b η_ab                                                                 (13")

        G_μ^a = G_μν q_b^ν η^ab                                                                 (13''')

        T_μ^a = T_μν q_b^ν η^ab                                                                 (14')

        q_μ^a = g_μν q_b^ν η^ab                                                                 (15')

        q_a^μ = g^μν q_ν^b η_ab                                                                 (15'')

In order to derive Eqs. (4) and (5), start from the Evans field Eq. (1) and use the following relations:

        G_μ^a = − ¼ R q_μ^a ,                                                                 (16)

        T_μ^a = ¼ T q_μ^a .                                                                 (17)

Equations (16) and (17) are derived from the definitions [1-12] of R and T introduced originally by Einstein [12]:

        R = g^μν R_μν ,         T = g^μν T_μν .                                                 (18)

Using the Einstein convention [12]

        g^μνg_μν = 4                                                                                 (19)

and the Cartan convention [11]

        q_μ^a q_a^μ = 1,                                                                                                 (20)

Unknown and wrong "convention"! Correct is

        q_μ^a q_a^μ = 4 ,                                                                                                 (20')

not due to Cartan but due to the fact that the matrices (q_μ^a) and (q_a^μ) are mutual inverses and Equ. (20) gives the trace of the 4×4 unit matrix.

together with the definitions (13) and (14), we obtain:

        R = g^μν R_μν

        = q_a^μ q_b^ν η^ab R_μ^a q_ν^b η_ab

        = (η^abη_ab)(q_b^ν q_ν^b) (q_a^μ R_μ^a) = 4 q_a^μ R_μ^a .                                                 (21)

This calculation is wrong due to total ignorance of the rules of tensor calculus.

The next erroneous step follows immediately:

Multiply either side of Eq. (21) by q_μ^a to obtain

        R_μ^a = ¼ Rq_μ^a ,                                                                                                 (22)

        G_μ^a = R_μ^a − ½ R q_μ^a = − ¼ R q_μ^a ,                                                                 (23)

which is Eq. (16). Similarly, we obtain Eq. (17).

This instruction "multiply" is not feasible since the indices μ and a are not free. The result, the eqs. (22) and (23), are obviously wrong since asserting the proportionality of the matrices (R_μ^a), (G_μ^a), (T_μ^a) to (q_μ^a), which is equivalent to the proportionality of the matrices (R_μν), (G_μν), (T_μν) to (g_μν). So there is no logical way to the aimed equations (22) and (23) which are identical with Eq.(16).

Herewith Evans' calculation has completely broken down.

Remark
On the basis of the equations (13)-(15) the Einstein field equations (11) are trivially equivalent to the Evans Field equations:

        R_μ^a − ½ R q_μ^a = k T_μ^a | · η_abq_ν^b     Û     R_μν − ½ R g_μν = k T_μν | · η^abq_b^ν.

. . .

The starting point for this class of wave equation are the Einsteinian definitions [12]

        R = R_μν g^μν ,                                                                                                 (46)

        T = T_μν g^μν .                                                                                                 (47)

Multiplication on both sides by g_μν gives

        R g_μν = 4 R_μν, T g_μν = 4 T_μν,                                                                 (48)

i.e.,

        R_μν = ¼ R g_μν, T_μν = ¼ T g_μν.                                                                 (49)