  The paper presents a classification of quadratic extension algebras, also
known as algebras of degree 2, as well as several characterizations of
quaternion algebras over a field (of characteristic not 2). The presentation is
not restricted to finite-dimensional algebras. The 'pure calculus' on a
quaternion algebra is introduced; it generalizes the 'vector calculus' of the
Hamilton's quaternions, and is instrumental in establishing, in a clear and
simple way, the isomorphism between the group consisting of all automorphisms
and all anti-automorphisms of a quaternion algebra and the orthogonal group of
the norm form on the subspace of pure quaternions. The paper concludes with a
glimpse of quaternion algebras over an integral domain (of characteristic not
2), and of their facility in studying ternary quadratic forms over an integral
domain.
