(This question only concerns tensors over finite-$n$-dimensional real vector spaces.)
The Kulkarni-Nomizu product is a multilinear map from of pairs of symmetric rank-2 tensors to the space of algebraic curvature tensors. For which dimensions is this map injective?
The set of pairs of symmetric rank-2 tensors has dimension $n(n+1)$. The set of algebraic curvature tensors has dimension $n^2(n-1)/12$. The former is greater for $n<4$, the latter is greater for $n>4$ (so the map cannot be surjective), and they are equal if $n = 4$.
I suspect that for $n < 4$, the Kulkarni-Nomizu product map is surjective but not injective, for $n = 4$ it is bijective, and for $n > 4$ it is injective but not surjective. Is this correct?