noncommutative algebra - Uniqueness of maximal subfields

I think this question presupposes something that is incorrect. Here is an example. Fix a field $F$ of characteristic 2 and put $Z$ for the field $F(a,b)$ where $a$ and $b$ are indeterminates. Define $D$ to be the quaternion division algebra with center $Z$ generated by elements $i, j$ satisfying

$$i^2 = a,quad j^2 + j = b,quad ij = (j+1)i.$$
Let us denote it by $lt a,b]$, as PK Draxl does in his book "Skew Fields". This is a division algebra because $a$ is not a norm from the separable quadratic extension $E$ obtained from $Z$ by adjoining a root of $x^2 + x + b$.

Obviously $D$ contains the purely inseparable quadratic extensions $Z(sqrt{a})$. I claim it also contains the extension $Z(sqrt{ab})$. To see this, we calculate in the Brauer group:

$$lt a,b] = lt ab^2,b] = lt ab,b] + lt b,b] = lt ab, b]$$
where the last equality is because $lt b,b]$ is split, i.e. isomorphic to 2-by-2 matrices, which follows from the fact that $b$ itself is a norm from $E$ (in fact, the norm of the element $x$).

Because $b$ is a nonsquare in $K$, we have found two non-isomorphic purely inseparable quadratic extensions in $D$ of exponent 1.

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