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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?

Mekanik
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    They are all correct. The difference is only convention, whether you like $H\rtimes G$ or you prefer $H\ltimes G$. The first one uses $H\oplus G$ as underlying set for the semidirect sum, instead of $H\times G$, or $G\times H$, for the semidirect product. – Dietrich Burde Sep 20 '16 at 20:05
  • But I am confused that in the first line (from wikipedia) the bracket takes pairs but the output is not a pair. – Mekanik Sep 22 '16 at 15:54
  • Yes, this is a bit sloppy. They identify $(a,b)$ in $A\oplus B$ then as an element $a+b$ in $A+B$. – Dietrich Burde Sep 22 '16 at 19:20

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