Manin triple
In mathematics, a Manin triple ( g , p , q ) {\displaystyle ({\mathfrak {g}},{\mathfrak {p}},{\mathfrak {q}})} consists of a Lie algebra g {\displaystyle {\mathfrak {g}}} with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras p {\displaystyle {\mathfrak {p}}} and q {\displaystyle {\mathfrak {q}}} such that g {\displaystyle {\mathfrak {g}}} is the direct sum of p {\displaystyle {\mathfrak {p}}} and q {\displaystyle {\mathfrak {q}}} as a vector space. A closely related concept is the (classical) Drinfeld double, which is an even dimensional Lie algebra which admits a Manin decomposition.