Weak Leibniz Algebras

An algebra with the identities $$[a,b]c=2a(bc)-2b(ac)$$ and $$a[b,c]=2(ab)c-2(ac)b$$ is called a weak Leibniz algebra. It is shown that any weak Leibniz operad is self-dual and not Koszul. It is also proved that the polarization of any weak Leibniz algebra is a transposed Poisson algebra and vice versa, the depolarization of any transposed Poisson algebra is a weak Leibniz algebra.