Indeed, Evans' exterior "covariant derivative" DÙ defined in [2,(17.16)] by
is not covariant. The simplest example is the group SU(2) with its bi-invariant metric and basis of left-invariant vector fields.
The curvature tensor can be easily computed:
Rijkl = ½ (δilδjk − δik δjl)
The connection coefficients are also easily calculated:
ωikj = ½ εikj
(cf. for instance Coquereaux and Jadczyk, "Riemannian geometry, fiber bundles, Kaluza-Klein theories and all that", World Scientific, 1988, pp. 10-13)
Therefore ωcb Ù Rac ≠ 0 and it is evidently not a horizontal form - in other words: it does not define a tensorial object or, as physicists use to say: it is not gauge-invariant. Therefore (D') does not define a "covariant" operation.
Evans asserts in (4.8) of his GCUFT book (vol.1)
To see that the "covariant derivative of the index contracted Christoffel symbol" is not covariant, it is enough to consider the simplest 1-dimensional case. Let the manifold M be the half-line M := (0,∞). We endow it with natural coordinate x in (0,∞) and the metric tensor g11(x) := 1 for all x in (0,∞). This is a flat metric, and the only Christoffel symbol in one dimension, Γ111, vanishes identically:
Γ111 = 0, for all x in (0,∞).
In particular Evans' D1 Γ111 = 0 identically.
Since D1 Γ111 is supposedly a scalar, it should vanish identically in any other coordinate system.
Let now x' be another coordinate, taking values in R given by x'(x) := log x , the inverse transformation is x(x') = ex'. We can easily compute the metric tensor component in the primed system:
g1'1'(x') = [∂ x/∂x']2 = e2x' .
This gives us the Christoffel symbol in the primed coordinate system:
Γ1'1'1' = ½ g1'1' ∂g1'1'/∂x' = 1 , for all x' in R.
Therefore Evans' covariant derivative is now:
D1' Γ1'1'1' = ∂Γ1'1'1'/∂x' + Γ1'1'1' Γ1'1'1' = g1'1' Γ1'1'1' Γ1'1'1' = e−2x' · 1² = e−2x' ≠ 0
for all x'. In particular at x'=0 (x=1) we get Evan's "covariant derivative" equal 1 while, if believed to be covariant, it should have value 0.
Remarkable, that Evans' assertion could pass the refereeing of a prestigious journal.
 M.W. Evans, A Generally Covariant Wave Equation for Grand Unified Field Theory,
Foundations of Physics Letters, Vol. 16, No. 6, Dec. 2003, pp. 513-547(35), Springer
 M.W. Evans, GENERALLY COVARIANT UNIFIED FIELD THEORY, Arima 2006