I am currently working through the classification of finite subgroups of $\text{SL}_2(\mathbb{C})$ as done in Klaus Lamotke's book on "Regular Solids and Isolated Singularities" (pages 33 ff.).
In this a group is a binary subgroup $G \subseteq \text{SL}_2(\mathbb{C})$ if there is a $\Gamma \subseteq \text{SO}(3)$ s.th. G is conjugate to the preimage of this $\Gamma$ under the epimorphism $\rho: \text{SU}(2) \to \text{SO}(3)$.
In the book it is simply stated that $G$ is binary if and only if the negative identity matrix is in $G$.
For me it is clear why $G$ binary implies $-E \in G$ (using that $-E$ is the only element of order 2 in $\text{SL}_2(\mathbb{C})$. However I cannot find a way to prove the converse.
