I'm reading Serre's A course in Arithmetic and there's one part in his proof that I don't understand. In his proof of Ch.4, n.2.2, Thm 6, he's trying to show that
$$(a_1,a_2)(a_3,a_4)(-1,a_3a_4)=-(-1,-1).$$
I could show this myself but I couldn't follow Serre's proof. Here's the part I don't understand:
$$(a_1,a_2)(a_3,a_4)(-1,a_3a_4)=(a_1,a_2)(-a_3,a_4)(-1,-1)$$
When I was trying to follow his proof, here's what I got.
\begin{align*} (a_1,a_2)(a_3,a_4)(-1,a_3a_4)&=(a_1,a_2)(a_3,a_4)(-1,a_4)(-1,a_3)\\ &=(a_1,a_2)(-a_3,a_4)(-1,a_3). \end{align*}
So now, doesn't it mean $(a_3,a_3)=(-1,a_3)=(-1,-1)$? I'm so confused how this can work. Can anyone help me out?
(Just to be clear, I don't want to see the proof of the theorem itself; as I mentioned, I managed to prove it myself. I just want to follow Serre's proof.)