  We prove a coordinatization theorem for unital alternative algebras
containing 2 x 2 matrix algebra with the same identity element 1. This solves
an old problem announced by Nathan Jacobson on the description of alternative
algebras containing a generalized quaternion algebra H with the same 1, for the
case when the algebra H is split. In particular, this is the case when the
basic field is finite or algebraically closed.
