My question is the following :

Given an algebraic curvature operator $R\in S^2_B(\Lambda^2\mathbb{R}^n)$, is there an a simple criterion to know if this curvature operator can occur as the curvature operator of symmetric space ?

I would be almost equally happy if someone can point to a way to know if $R$ can be the curvature operator of some Riemannian manifold with reduced holonomy $G\subset SO(n,\mathbb{R})$.

I suspect this might be linked to the fact that $\Lambda^2\mathbb{R}^n=\mathfrak{so}(n,\mathbb{R})$ and how $R$ acts on $\mathfrak{g}\subset\mathfrak{so}(n,\mathbb{R})$. So far, the only link I four is that the image of $R$ has to be contained in $\mathfrak{g}$. But is it possible to write a condition NOT along the line of "There is a Lie subalgebra $\mathfrak{g}\subset\mathfrak{so}(n,\mathbb{R})$ such that..."

My question is the following :

Given an algebraic curvature operator $R\in S^2_B(\Lambda^2\mathbb{R}^n)$, is there an a simple criterion to know if this curvature operator can occur as the curvature operator of symmetric space ?

I would be almost equally happy if someone can point to a way to know if $R$ can be the curvature operator of some Riemannian manifold with reduced holonomy $G\subset SO(n,\mathbb{R})$.

I suspect this might be linked to the fact that $\Lambda^2\mathbb{R}^n=\mathfrak{so}(n,\mathbb{R})$ and how $R$ acts on $\mathfrak{g}\subset\mathfrak{so}(n,\mathbb{R})$. So far, the only link I found is that the image of $R$ has to be contained in $\mathfrak{g}$. I have no idea if this is sufficient. And I wonder if it is possible to write a condition NOT along the line of "There is a Lie subalgebra $\mathfrak{g}\subset\mathfrak{so}(n,\mathbb{R})$ such that..."