Let $K$ be a field of characteristic $0$, $D$ a central semi-simple division algebra of dimension $d^2$ over $K$, and $n$ a positive integer. Let $R$ be a maximal subfield of $M_n(D)$, then is $\dim_K R$ $\le nd$? I know, if $n=1$ or if $D$ is simple algebra then it is true. How about the general case?
I want to show it for $ D := \operatorname{End}^0 X$, where $X$ is a simple abelian variety over an algebraically closed field (of characteristic not zero).
Note, if such $D$ is of type I, II or III (see for example Mumford's Abelian varieties, page 202), then since $D$ is a quarternion or a field, we have the result. But in the case of type IV, $D$ maybe non-simple.