On the subspace $L((x\land y)^m)$ of $S^m(\land^2\mathbb R^4)$

We take the exterior power $\mathbb R^4\land\mathbb R^4$ of the space $\mathbb R^4$, its $m$th symmetric power $V=S^m(\land^2\mathbb R^4)=(\mathbb R^4\land\mathbb R^4)\vee(\mathbb R^4\land\mathbb R^4)\vee\cdots\vee(\mathbb R^4\land\mathbb R^4)$, and put $V_0=L((x\land y)\vee\cdots\vee(x\land y)\colon x,y\in\mathbb R^4)$. We find the dimension of $V_0$ and an algorithm for distinguishing a basis for $V_0$ efficiently. This problem arose in vector tomography for the purpose of reconstructing the solenoidal part of a symmetric tensor field.