I am learing how to solve the system of equations
\begin{align} r_{12}+r_{21}=t,\ [[r,r]]=0, \end{align} where $t$ is the Casimir element of $g\otimes g$ corresponding to a non-degenerate invariant bilianear form $(,)$ on $g$.
Let $g$ be a Lie algebra, $\prod$ the set of simple roots of $g$, $\tau$ a bijection $\prod_{1}\rightarrow\prod_{0} $, where $\prod_{1},\prod_{0} \subset \prod$. For a given admissible triple $(\prod_{1},\prod_{0},\tau)$,
I want to know why $r_{12}^{0}+r_{21}^{0}=t_{0}$ is equivalent to $r_{\alpha \beta}^{0}+r_{ \beta\alpha}^{0}=(\alpha, \beta)$, where $r_{\alpha \beta}^{0}=(\alpha \otimes \beta)(r^{0})$, $r^{0}\in \mathfrak{h}\otimes\mathfrak{h}$.
