Let $C$ be the ideal class group of $\mathbb{Q}(\sqrt{-6})$. I already showed that the ideal $(2,\sqrt{-6})\in C$ is not principal in $\mathbb{Q}(\sqrt{-6})$, but it is principal in $\mathbb{Q}(\sqrt{-6},\sqrt{2})$ (since $(2,\sqrt{-6})=(\sqrt{2})$).
My problem is to show that the ideal $(2,\sqrt{-6})$ respresents a class in
$A^-$, where $A^-$ is defined by the following:
$A$ is the $2$-Sylow subgroup of $C$, $A^\pm=\{x\in A|J(x)=x^{\pm 1}\}$, where $J$ is the complex conjugation.
(exercise 13.3 (c) of Washington's Introduction to Cyclotomic Fields)
I found out that the class number of $\mathbb{Q}(\sqrt{-6})$ is $2$, so the ring of integers $\mathbb{Z}(\sqrt{-6})=(1)$ and $(2,\sqrt{-6})$ represent the only ideal classes. Let $I$ be an ideal. Then $I\in A^+\cap A^-$ means that $I^{-1}=J(I)=I$, which is equivalent to $I^2=1$ (so $I^2$ is a principal ideal). Since $(2,\sqrt{-6})^2=(2)$, $(2,\sqrt{-6})\in A^+\cap A^-$, in particular $(2,\sqrt{-6})\in A^-$.