I came across the following theorem:
Let $\mathcal{L}$ be a Lie algebra without non-trivial $\frac{1}{2}$-derivations. Then every transposed Poisson structure defined on $\mathcal{L}$ is trivial.
If $\mathcal{L}$ is a Lie algebra without non-trivial, then left-multiplication by a scalar $\lambda$ won't be a $\frac{1}{2}$-derivation, $\lambda \left[x ; y\right] \neq \frac{1}{2}\left[\lambda x ; y\right] + \frac{1}{2}\left[x ; \lambda y\right]$, but I don't know how to arrive at that every Poisson structure should be trivial and even if the last inequality could be considered. Any suggestions