TITLE: Inner forms of $GL(2)$
QUESTION [5 upvotes]: I know that inner forms of $GL(2)$ are quaternion algebras. However, I cannot find the proof myself. 
First, since quaternion algebras are forms of $GL(2)$ by the usual embedding in matrices, they are automatically inner by Skolem-Noether theorem.
But how can I prove the converse, that is to say if I have a group "inner isomorphic" (i.e. by conjugation) to $GL(2)$ over an algebraic closure, then it is a quaternion algebra?
Thanks for any clue or reference!

REPLY [9 votes]: If you have a group $G$ over $k$ such that $G_{\overline{k}}$ is isomorphic to $GL_2$ and, and each element of $\operatorname{Gal}(\overline{k}|k)$ acts on this $GL_2$ by the standard action times conjugation by some (unique up to scalars element), then we can use the same cocycle to form a twisting of the algebra $M_2(k)$, because $PGL_2$ acts by conjugation on $M_2$.
By Hilbert 90, the twisted form of $M_2(K)$ remains a $4$-dimensional vector space, with algebra structure. Any two-sided ideal must remain a two-sided ideal over $\overline{k}$, and similarly with central elements, so it is a (rank 4) central simple algebra, i.e. a quaternion algebra.
Because the group of units of this algebra is $GL_2$, and this restriction is compatible with the action of $PGL_2$ and Galois, so the multiplicative group of this algebra is a twisted form of $GL_2$.