Defining Bianchi identities for expressions w/o the Levi-Civita tensor

[Sukruti]

Hello,

Any suggestions for how to define the two Bianchi identities shown in the attachment, in order to implement them on expressions that do not consist of the Levi-Civita tensor?

RiemannSimplify can implement the Bianchi identities but it works only for expressions which are exclusively in terms of the curvature tensors/scalar. I need to use the identities for expressions consisting of derivatives of a scalar field along with the curvature tensors/scalar, so I'm looking for a way to define the two Bianchi identities.

Best,

Sukruti