Let $k$ be a field of characteristic 0 and let $\varphi:\mathfrak{g}\rightarrow\mathfrak{f}$ and $\psi:\mathfrak{h}\rightarrow\mathfrak{f}$ be maps of Lie algebras. Is there a reference showing that the pullback (in the category of Lie $k$-algebras) of $\mathfrak{g}$ and $\mathfrak{h}$ along these maps is given by the following formula



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> $\mathfrak{g}\times\_\mathfrak{f}\mathfrak{h}=\{(x,y)\in\mathfrak{g}\times\mathfrak{h}\mid \varphi(x)=\psi(y)\}$
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> 
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EDIT: Just to be extra clear. I do know how to prove it, but I cannot find a place in the literature where this is proved or stated and I was wondering if you knew of one.



Pullback of Lie algebras
A reference for pullbacks for modules over a ring: Proposition 5.11, p.222 in "An Introduction to Homological Algebra" by Rotman. For Lie $k$-algebras (or just $k$-algebras), you could refer to this and say the same construction/proof works.


