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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 this MO answerthis MO answer.

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 this MO answer.

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 this MO answer.

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Abdelmalek Abdesselam
Abdelmalek Abdesselam
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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 this MO answer.