  Let $\mathfrak{g}$ be the derived subalgebra of a Kac-Moody Lie algebra of
finite type or affine type, $\mu$ a diagram automorphism of $\mathfrak{g}$ and
$L(\mathfrak{g},\mu)$ the loop algebra of $\mathfrak{g}$ associated to $\mu$.
In this paper, by using the vertex algebra technique, we provide a general
construction of current type presentations for the universal central extension
$\widehat{\mathfrak{g}}[\mu]$ of $L(\mathfrak{g},\mu)$. The construction
contains the classical limit of Drinfeld's new realization for (twisted and
untwisted) quantum affine algebras ([Dr]) and the Moody-Rao-Yokonuma
presentation for toroidal Lie algebras ([MRY]) as special examples. As an
application, when $\mathfrak{g}$ is of simply-laced type, we prove that the
classical limit of the $\mu$-twisted quantum affinization of the quantum
Kac-Moody algebra associated to $\mathfrak{g}$ introduced in [CJKT1] is the
universal enveloping algebra of $\widehat{\mathfrak{g}}[\mu]$.
