In the book Algebra IX: Finite Groups of Lie type and Finite Dimensional Algebra, the authors Kostrikin-Shafarevich mention (p. 159) that
If $A$ is a central simple algebra over $F$ of finite dimension and if $K$ is a maximal subfield of $A$, then $K$ may not be Galois extension of $F$.
On the other hand, in a paper on Central Simple Algebras, Rowen mentiones
Every division algebra of degree $2, 3, 4, 6$, or $12$ is a crossed product (i.e. maximal subfields are Galois extensions; am I right? Here degree of $D$ over $F$ is $n$ means $\dim_F(D)=n^2$.)
Q. What is example of a non-commutative simple (or division) algebra of finite dimension over $F$ in which a maximal subfield is not Galois extension of $F$?
(I have no idea of how difficult is this problem in the literature, and I have never seen simple algebras other than quaternions and matrix algebras over quaternions or fields.)