Anticommutator of gauge covariant derivatives

I must convert some dimension-6 operators I’ve obtained to the SILH base (ref: this, "Review of the SILH basis", CERN presentation by R. Contino).
In this conversion I’ve got operators such as $D_{mu}D_{nu}H^{dagger} D^{nu}D^{mu}H = frac{1}{4}(G’_{munu}H^{dagger} G^{primenumu}H + {D_{mu},D_{nu}} H^{dagger} {D_{nu},D_{mu}}H)$.
In the SILH base I should have only operators of the kind $DG DH^{dagger} H, GGH ^{dagger} H $, i.e. no second covariant derivatives on the Higgs doublet.
However, I’m having problems in re-writing ${D_{mu}, D_{nu}}H$. What is this anticommutator equivalent to?
I’m not finding anything in literature, and opening this derivative explicitely breaks the covariance of the operator.

(Note: $G’ = frac{g^*}{i}G$)