Quasi-Frobenius Lie algebra

In mathematics, a quasi-Frobenius Lie algebra 
over a field formula_2 is a Lie algebra 
equipped with a nondegenerate skew-symmetric bilinear form 
for all formula_8, formula_9, formula_10 in formula_5.
If formula_12 is a coboundary, which means that there exists a linear form formula_13 such that
then
is called a Frobenius Lie algebra.
Equivalence with pre-Lie algebras with nondegenerate invariant skew-symmetric bilinear form.
If formula_1 is a quasi-Frobenius Lie algebra, one can define on formula_5 another bilinear product formula_18 by the formula
Then one has
formula_20 and 
is a pre-Lie algebra.