On the definition of the dual Lie coalgebra of a Lie algebra

Let $L$ be a Lie algebra over a field $K$. The dual Lie coalgebra $L^\circ$ of $L$ has been defined by W. Michaelis to be the sum of all good subspaces $V$ of the dual space $L^*$ of $L$: $V$ is good if ${}^tm (V) \subset V \otimes V$, where $m$ is the multiplication of $L$. We show that $ L^\circ = {}^t m^{-1} ( L^* \otimes L^* )$ as in the associative case.