ag.algebraic geometry - Division Algebras as Algebraic Groups

Suppose D splits over a finite extension K/F, i.e., the tensor product of D with K over F is isomorphic to M_n(K). Then D^x is the group of F-points of an algebraic group over F that exists as a direct factor (along with all other F-division algebras that split over K, and GL_n,F) in the restriction of scalars Res^K_F GL_n,K.

I don't know an explicit presentation in general (say, starting from a Brauer class), although if K/F is a cyclic Galois extension, there is a nice cyclic algebra construction. I think more details can be found in Serre's Local Fields and Cornell-Silverman.