  We introduce the notion of $\lambda$-double Lie algebra, which coincides with
usual double Lie algebra when $\lambda = 0$. We state that every
$\lambda$-double Lie algebra for $\lambda\neq0$ provides the structure of
modified double Poisson algebra on the free associative algebra. In particular,
it confirms the conjecture of S. Arthamonov (2017). We prove that there are no
simple finite-dimensional $\lambda$-double Lie algebras.
