The universal central extension of the three-point $\mathfrak{sl}_2$ loop algebra

We consider the three-point loop algebra,

$L= \mathfrak{sl}_2\otimes \mathbb{K} \lbrack t, t^{-1}, (t-1)^{-1}\rbrack,$

where $\mathbb{K}$ denotes a field of characteristic 0 and $t$ is an indeterminate. The universal central extension $\widehat L$ of $L$ was determined by Bremner. In this note, we give a presentation for $\widehat L$ via generators and relations, which highlights a certain symmetry over the alternating group $A_4$. To obtain our presentation of $\widehat L$, we use the realization of $L$ as the tetrahedron Lie algebra.