I am trying to understand the dual Lie algebra in a Lie bialgebra.
In the above article, it is said that:
"Let ${\displaystyle {\mathfrak {g}}}$ be any semisimple Lie algebra. To specify a Lie bialgebra structure we thus need to specify a compatible Lie algebra structure on the dual vector space. Choose a Cartan subalgebra ${\displaystyle {\mathfrak {t}}\subset {\mathfrak {g}}}$ and a choice of positive roots. Let ${\displaystyle {\mathfrak {b}}_{\pm }\subset {\mathfrak {g}}}$ be the corresponding opposite Borel subalgebras, so that ${\displaystyle {\mathfrak {t}}={\mathfrak {b}}_{-}\cap {\mathfrak {b}}_{+}}$ and there is a natural projection ${\displaystyle \pi :{\mathfrak {b}}_{\pm }\to {\mathfrak {t}}}$. Then define a Lie algebra $$ {\displaystyle {\mathfrak {g'}}:=\{(X_{-},X_{+})\in {\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}\ {\bigl \vert }\ \pi (X_{-})+\pi (X_{+})=0\}}$$ which is a subalgebra of the product ${\displaystyle {\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}}$, and has the same dimension as ${\displaystyle {\mathfrak {g}}}$. Now identify ${\displaystyle {\mathfrak {g'}}}$ with dual of ${\displaystyle {\mathfrak {g}}}$ via the pairing $$ {\displaystyle \langle (X_{-},X_{+}),Y\rangle :=K(X_{+}-X_{-},Y)}$$ where ${\displaystyle Y\in {\mathfrak {g}}}$ and $K$ is the Killing form. This defines a Lie bialgebra structure on ${\displaystyle {\mathfrak {g}}}$, and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group. "
Where does the condition $\pi(X_-)+\pi(X_+)=0$ come from? Thank you very much.