Consider the affine Kac-Moody algebra $\mathfrak g=\widehat{\mathfrak{sl}}_r(\mathbb C((t)))$ and consider the two involutions $$a(t)\rightarrow \sigma(a(t))=-\,^ta(-t),$$ and when $r$ is even $$a(t)\rightarrow \tau(a(t))=-J_r\,^ta(-t)J_r^{-1},$$ where $$J_r=\begin{pmatrix} 0& D_r \\ -D_r& 0\end{pmatrix},$$ where $D_r$ is the anti-diagonal matrix with all entries equal $1$.

What is the normalized $2-$cocycle defining the twisted Kac-Moody algebra $\mathcal L(\mathfrak{sl}_r,\sigma)$ and $\mathcal L(\mathfrak{sl}_r,\tau)$? In other words, what is the canonical central element of these algebras w.r.t. that of $\mathfrak{g}$?

The expectation is that they are the same for $\sigma$ case, and it is half that of that of $\mathfrak{g}$ in the othercase. Thanks