We take the exterior power ℝ4 ∧ ℝ4 of the space ℝ4, its mth symmetric power V = S m(∧2ℝ4) = (ℝ4 ∧ ℝ4) ∨ (ℝ4 ∧ ℝ4) ∨ ... ∨(ℝ4 ∧ ℝ4), and put V0 = L(( x ∧ y)∨ ... ∨( x ∧ y): x, y ∈ ℝ4). We find the dimension of V0 and an algorithm for distinguishing a basis for V0 efficiently. This problem arose in vector tomography for the purpose of reconstructing the solenoidal part of a symmetric tensor field.