So that isn’t particularly exciting or New… but there's more interesting things we have when have a inner product defined on that symmetric, non-degenerate and invariant (Killing semi-simple Lie algebra, for instance); against; against a invariant inner product on contrary; Notice about 3-form gives us . This 3-form is cyclically symmetric, because Killing is invarianI Lie is on .
So that isn’t particularly exciting or new… but there are more interesting things availible when we have a inner product defined on that is symmetric, non-degenerate and invariant (the Killing form on a semi-simple Lie algebra, for instance). An invariant inner product on is one such that . Notice this gives us a nice 3-form . This 3-form is cyclically symmetric, because .