Another elementary description of the degree four invariant is as
$$
\epsilon_{i_1i_2i_3i_4i_5i_6}\ \epsilon_{i_7i_8i_9i_{10}i_{11}i_{12}}
\ \omega_{i_1i_2i_3\ }\ \omega_{i_4i_5i_7}\ \omega_{i_6i_8i_9}\ \omega_{i_{10}i_{11}i_{12}}
$$
where indices are summed from 1 to 6. The epsilon notation is as in
<a href="http://mathoverflow.net/questions/255492/how-to-constructively-combinatorially-prove-schur-weyl-duality/255853#255853">this MO answer</a>.