Twisted toroidal Lie algebras and Moody-Rao-Yokonuma presentation

Let $\text{fg}$ be an affine Kac-Moody algebra, and $\mu$ a diagram automorphism of $\text{fg}$. In this paper, we give an explicit realization for the universal central extension $\text{wh}\text{fg}[\mu ]$ of the twisted loop algebra of $\text{fg}$ related to $\mu$, which provides a Moody-Rao-Yokonuma presentation for the algebra $\text{wh}\text{fg}[\mu ]$ when $\mu$ is non-transitive, and the presentation is indeed related to the quantization of toroidal Lie algebras.