Show that the center of quaternions group $\textit{Q}$ is generated by the unique element with order 2.

Question

$\textit{Q}$ is a group with order $8$, generated by $a,b$ where $a^4=1$, $b^2=a^2$ and $bab^{-1}=a^{-1}$.

I already proved that the unique element of $\textit{Q}$ with order $2$ is $a^2$.

How can I prove that $Z(\textit{Q})=\left<a^2\right>$?