Where does the condition $\pi(X_-)+\pi(X_+)=0$ in dual Lie algebra come from?

I am trying to understand the dual Lie algebra in a Lie bialgebra.

In the above article, it is said that:

"Let ${\mathfrak {g}}$ 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 ${\mathfrak {t}}\subset {\mathfrak {g}}$ and a choice of positive roots. Let ${\mathfrak {b}}_{\pm }\subset {\mathfrak {g}}$ be the corresponding opposite Borel subalgebras, so that ${\mathfrak {t}}={\mathfrak {b}}_{-}\cap {\mathfrak {b}}_{+}$ and there is a natural projection $\pi :{\mathfrak {b}}_{\pm }\to {\mathfrak {t}}$. Then define a Lie algebra $${\mathfrak {g'}}:=\{(X_{-},X_{+})\in {\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}\ {\bigl \vert }\ \pi (X_{-})+\pi (X_{+})=0\}$$ which is a subalgebra of the product ${\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}$, and has the same dimension as ${\mathfrak {g}}$. Now identify ${\mathfrak {g'}}$ with dual of ${\mathfrak {g}}$ via the pairing $$\langle (X_{-},X_{+}),Y\rangle :=K(X_{+}-X_{-},Y)$$ where $Y\in {\mathfrak {g}}$ and $K$ is the Killing form. This defines a Lie bialgebra structure on ${\mathfrak {g}}$, 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.

• 瞎猜的, 如果定义换成 $$\mathfrak {g'}:=\{(X_{-},X_{+})\in {\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}\ {\bigl \vert }\ \pi (X_{-})-\pi (X_{+})=0\}$$和$$\langle (X_{-},X_{+}),Y\rangle :=K(X_{+}+X_{-},Y),$$是不是还是成立?所以这个条件只是用来限制维数? – Nirvanacs Dec 12 '17 at 0:27
• @Nirvanacs, thank you very much for your comments. – LJR Dec 12 '17 at 1:03