Non-Abelian Order 8 Group with One Order 2 Element is Quaternion Group/Lemma 2

Lemma

 * $\left({\pm a}\right)^2 = \left({\pm b}\right)^2 = \left({\pm c}\right)^2 = -1$

Proof
WLOG, only $a$ is checked. The proofs for other $5$ elements are similar.

So $a^2 = 1$ or $-1$.

As order of $a=4$, $a^2 \ne 1$.

$a^2 = -1$