Showing that a quadratic extension of $\mathbb{Q}$ is contained in $\mathbb{R}$

Question

Suppose that L/$\mathbb{Q}$ is a Galois extension such that $Gal(L/\mathbb{Q}) \cong Q_8$ and that $K/\mathbb{Q}$ is a quadratic extension with $\mathbb{Q} \subset K \subset L$ I am trying to show that $K \subset \mathbb{R}$ and so far I've only showed that every sub extension of $L/\mathbb{Q}$ is a Galois extension. I am thinking that it must be something related to complex conjugation but I am not seeing it.

Answer 1

As suggested by JSE here -> https://mathoverflow.net/a/39529, note that the element of order 2 in the quaternion group is in the kernel of every homomorphism from $Q_8 \rightarrow \Bbb{Z}/2\Bbb{Z}$ which is to say the element of order 2 fixes each of the quadratic subfields.