$\mathbb{Q}(i,\sqrt{2})$ is a splitting field of $f_{\mathbb{Q}}^{i+\sqrt{2}} $ and $\operatorname{Aut}(\mathbb{Q}(i,\sqrt{2})) \cong V_{4}$

Question

So hi all, I'm trying to show that $\operatorname{Aut}(\mathbb{Q}(i,\sqrt{2}))$ is isomorphic to $V_4$ but I don't have enough intuition for this yet to solve it. Would you like to give me some advice? Thanks in advance. And sorry if someone else asked this question already; I couldn't find it.

What I've found already is that $f_{\mathbb{Q}}^{i+\sqrt{2}} = X^{4} - 2X^{2} + 9$ and I think that this yields that $\mathbb{Q}(-i-\sqrt{2},-i+\sqrt{2},i-\sqrt{2},i+\sqrt{2}) = \mathbb{Q}(i,\sqrt{2}). $