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Product and semidirect product

For two Lie algebras g {\displaystyle {\mathfrak {g}}} and g ′ {\displaystyle {\mathfrak {g'}}}, the product Lie algebra is the vector space g × g ′ {\displaystyle {\mathfrak {g}}\times {\mathfrak {g'}}} consisting of all ordered pairs (x, x ′), x ∈ g, x ′ ∈ g ′ {\displaystyle (x,x'),\,x\in {\mathfrak {g}},\ x'\in {\mathfrak {g'}}}, with Lie bracket

This is the product in the category of Lie algebras. Note that the copies of g {\displaystyle {\mathfrak {g}}} and g ′ {\displaystyle {\mathfrak {g}}'} in g × g ′ {\displaystyle {\mathfrak {g}}\times {\mathfrak {g'}}} commute with each other: = 0. {\displaystyle =0.}

Let g {\displaystyle {\mathfrak {g}}} be a Lie algebra and i {\displaystyle {\mathfrak {i}}} an ideal of g {\displaystyle {\mathfrak {g}}}. If the canonical map g → g / i {\displaystyle {\mathfrak {g}}\to {\mathfrak {g}}/{\mathfrak {i}}} splits (i.e., admits a section g / i → g {\displaystyle {\mathfrak {g}}/{\mathfrak {i}}\to {\mathfrak {g}}}, as a homomorphism of Lie algebras), then g {\displaystyle {\mathfrak {g}}} is said to be a semidirect product of i {\displaystyle {\mathfrak {i}}} and g / i {\displaystyle {\mathfrak {g}}/{\mathfrak {i}}}, g = g / i ⋉ i {\displaystyle {\mathfrak {g}}={\mathfrak {g}}/{\mathfrak {i}}\ltimes {\mathfrak {i}}}. See also semidirect sum of Lie algebras.