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In the paper (Osaka J. Math. {\bf 46}: 403-409, 2009), Yang conjectured that a non-elementary subgroup $G$ of $SL(2, \bc)$ containing elliptic elements is discrete if for each elliptic element $g\in G$ the group $ $ is discrete, where $f\in SL(2,\bc)$ is a test map which is loxodromic or elliptic. The purpose of this paper is to give an affirmative answer to this question.

Discreteness criterion in $\mathrm{SL}(2,\mathbb{C})$ by a test map