Theorem: If $A$ is a finite-dimensional central simple algebra over a field $F$ and $B$ is a simple subalgebra of $A$, then $C_A(C_A(B))=B$.
In general (means, without condition on $B$, the relation $\supseteq$ holds in above.
Simplest (or trivial) example could be $B=A$ (am I right?). Then $C_A(A)=Z(A)$ and $C_A(Z(A))=A$.
(1) Can one give an elementary example of algebra $A$ and sub-algebra $B$ where strict inequality $\supset$ holds?
(2) Can one mention interesting applications of this theorem? (One may post a link also).
1) $A=\mathbb C$, $B=\mathbb R$
2) you can see a few simple applications at the end of this doc regarding Davison rings and the Brauer group.