  The $q$-Onsager algebra $O_q$ is defined by two generators and two relations,
called the $q$-Dolan/Grady relations. We investigate the alternating central
extension $\mathcal O_q$ of $O_q$. The algebra $\mathcal O_q$ was introduced by
Baseilhac and Koizumi, who called it the current algebra of $O_q$. Recently
Baseilhac and Shigechi gave a presentation of $\mathcal O_q$ by generators and
relations. The presentation is attractive, but the multitude of generators and
relations makes the presentation unwieldy. In this paper we obtain a
presentation of $\mathcal O_q$ that involves a subset of the original set of
generators and a very manageable set of relations. We call this presentation
the compact presentation of $\mathcal O_q$. This presentation resembles the
compact presentation of the alternating central extension for the positive part
of $U_q(\widehat{\mathfrak{sl}}_2)$.
