Riemann curvature space

Definition

Given data

A real vector space $V$ with an inner product $g$ (making it into the usual Euclidean space).

Definition part

The Riemann curvature space over $V$, denoted $RC(V)$, is the space of multilinear forms on $V$ in 4 variables, satisfying the following identities:

1. $R(X,Y,Z,T)=-R(Y,X,Z,T)$ (viz $R$ is antisymmetric in the first two variables)
2. $R(X,Y,Z,T)=-R(X,Y,T,Z)$ (viz $R$ is antisymmetric in the last two variables)
3. $R(X,Y,Z,T)+R(Y,Z,X,T)+R(Z,X,Y,T)=0$ (The first Bianchi identity)
4. $R(X,Y,Z,T)=R(Z,T,X,Y)$ (this follows from the above three)

Note that $RC(V)$ sits naturally as a subspace inside the space $Sym^{2}(Alt^{2}(V))$ (follows from the facts (1), (2) and (4).