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
, $a$ is checked.

The proofs for other $5$ elements are similar.

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

As the order of $a = 4$:
 * $a^2 \ne 1$

Hence:
 * $a^2 = -1$

as required.