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

Twisted toroidal Lie algebras and Moody-Rao-Yokonuma presentation