How to define a Tensor with the symmetries like Riemann tensor?

I want to define a tensor that has the first two symmetries of Riemann tensor or maybe the last two. The symmetries of Riemann tensor are: 1) $$R_{\alpha\beta\gamma\lambda}=R_{\gamma\lambda\alpha\beta}$$ 2)$$R_{\alpha\beta\gamma\lambda}=-R_{\alpha\beta\lambda\gamma}$$ and $$R_{\alpha\beta\gamma\lambda}=-R_{\alpha\beta\lambda\gamma}$$ 3)$$R_{\alpha \beta \gamma \lambda} +R_{\alpha\lambda\beta\gamma} + R_{\alpha\gamma\lambda \beta }=0$$ Could you help me to learn to define a tensor with mixed symmetries?

The skew symmetries and the interchange symmetry can be implemented easily by specifying them as symmetry generators in SymmetrizedArray:

rt=SymmetrizedArray[pos_:>Subscript["R",Row[pos]],{4,4,4,4},{
{{2,1,3,4},-1},(* Skew symmetry in first two slots *)
{{1,2,4,3},-1},(* Skew symmetry in last two slots *)
{{3,4,1,2},1}(* Interchange symmetry between first and last pair of slots*)
}]
SymmetrizedArrayRules[%]
Length[%]


returns

This seems to work fine: one can check the symmetries by looking at the components and the number of components is also correct: if I recall my General Relativity lectures correctly the Riemann tensor has 21 independent components in four dimensions (20 when including the first (algebraic) Bianchi identity $$R_{\alpha \beta \gamma \lambda} +R_{\alpha\lambda\beta\gamma} + R_{\alpha\gamma\lambda \beta }=0$$).

The first (algebraic) Bianchi identity can not be implemented easily in terms of symmetry generators but one could manually select a number of components ($$\frac{1}{24} d (d-1) (d-2) (d-3)$$ where $$d$$ is the dimension of the tensor (slots)) to be expressed in terms of two other components. Depending on your dimension $$d$$ this might make sense or not. In $$d<3$$ the identity gives no additional symmetry, in $$d=4$$ just one, and so it only looks profitable for larger $$d$$.