Verify the following relationship: $\nabla \cdot (a \times b) = b \cdot \nabla \times a - a \cdot \nabla \times b $
Use the Leibniz rule in the firts term:
$$ \frac{\partial}{\partial x}(S_aT_b - S_bT_a) = T_b\frac{\partial S_a}{\partial x} + S_a\frac{\partial T_b}{\partial x} - T_a\frac{\partial S_b}{\partial x} - S_b \frac{\partial Ta}{\partial x}. $$
Expanding the other two you should get the desired equality.
Another approach can be using index notation.