Bogomolov Multiplier of Lie algebras

In the work of Rostami et al., the Bogomolov multiplier of a Lie algebra $L$ over a field $\mathrm{\Omega }$ is defined as a particular factor of a subalgebra of the exterior product $L\wedge L$. If $L$ is finite dimensional, we identify this object as a certain subgroup of the second cohomology group $H^2(L,\mathrm{\Omega })$ by deriving a Hopf-Type formula. As an application, we affirmatively answer two questions posed by Kunyavskii regarding the invariance of the Bogomolov multiplier under isoclinism of Lie algebras and the existence of a family of Lie algebras with Bogomolov multipliers of unbounded dimension.