  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 central
extension $\wh\fg[\mu]$ 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 the
quantization of toroidal Lie algebras.
