Hello. I have some question on Cartan decomposition of unitary group, especially $U(2)$.

I am interested in local situation, that is p-adic or archimedian.

Let $F$ be a local field and $E$ be its quadratic extension. Let $V$ be a hyperbolic hermitian vector space of dimension $2$ and we consider $U(V)$.

Then by Cartan decomposition, (see Cartan decomposition of a unitary group?) we can decompose $U(V)=KM^{+}K$ where $M^+=${$x \in E^{\times}||x|\le1 $}.

Here, I am just curious whether the center of $U(V)$, that is just $U(1)$ by diagonal embedding into $U(V)$, is contained into $K$. Since $U(1)$ is compact, it seems that it is possible to take $K$ from the beginning to contain the center. Am I right?(As I don't know the exact shape of $K$, I am not sure it)

Any words or comments will be greatly appreciated.