Editing Lie algebra (section)

=== Product and semidirect product ===
For two Lie algebras <math>\mathfrak{g}</math> and <math>\mathfrak{g'}</math>, the ''[[direct product|product]]'' Lie algebra is the vector space <math>\mathfrak{g}\times \mathfrak{g'}</math> consisting of all ordered pairs <math>(x,x'), \,x\in\mathfrak{g}, \ x'\in\mathfrak{g'}</math>, with Lie bracket<ref>{{harvnb|Bourbaki|1989|loc=section I.1.1.}}</ref>
:<math> [(x,x'),(y,y')]=([x,y],[x',y']).</math>
This is the product in the [[product (category theory)|category]] of Lie algebras. Note that the copies of <math>\mathfrak g</math> and <math>\mathfrak g'</math> in <math>\mathfrak{g}\times \mathfrak{g'}</math> commute with each other: <math>[(x,0), (0,x')] = 0.</math>

Let <math>\mathfrak{g}</math> be a Lie algebra and <math>\mathfrak{i}</math> an ideal of <math>\mathfrak{g}</math>. If the canonical map <math>\mathfrak{g} \to \mathfrak{g}/\mathfrak{i}</math> splits (i.e., admits a section <math>\mathfrak{g}/\mathfrak{i}\to \mathfrak{g}</math>, as a homomorphism of Lie algebras), then <math>\mathfrak{g}</math> is said to be a [[semidirect product]] of <math>\mathfrak{i}</math> and <math>\mathfrak{g}/\mathfrak{i}</math>, <math>\mathfrak{g}=\mathfrak{g}/\mathfrak{i}\ltimes\mathfrak{i}</math>. See also [[Lie algebra extension#By semidirect sum|semidirect sum of Lie algebras]].