# Involutions of the second type in a division algebra

I'm trying to figure out some details about involutions of division algebra, thought maybe someone here might have a better insight.

Let $k$ be a $p$-adic or number field, and let $K=k[\sqrt{\delta}]$ be a non-trivial extension of degree $2$. For $x\in K$, let $\bar{x}$ denote the conjugate of $x$ under the non-trivial $K/k$ automorphism. Let $D$ be a division algebra of degree $\ell$ with $Z(D)=K$. For simplicity, we shall assume that $\ell$ is prime (or even $\ell=3$ is enough for the moment).