Just a small question about a possible typo in a wiki-article: In the article https://en.wikipedia.org/wiki/Lie_algebra_extension#Background_material under 'By semidirect sum', i assume what is defined in equation (7) is the external semidirect sum of two Lie algebras: $[(H,G),(H',G')]=[H,H']+[G,G']+\psi _{G}(H)-\psi _{{G'}}(H'),\quad H,H'\in {\mathfrak h},G,G'\in {\mathfrak g}$
But shouldnt it be written as:? $[(H,G),(H',G')]=([H,H']+\psi _{G}(H)-\psi _{{G'}}(H'),[G,G']),\quad H,H'\in {\mathfrak h},G,G'\in {\mathfrak g}$
or should it be $[(H,G),(H',G')]=([H,H']+\psi _{G}(H')-\psi _{{G'}}(H),[G,G']),\quad H,H'\in {\mathfrak h},G,G'\in {\mathfrak g}$
and what would be the difference between my second and third possibility?