  This paper is concerned with the description of exceptional simple Lie
algebras as octonionic analogues of the classical matrix Lie algebras. We
review the Tits-Freudenthal construction of the magic square, which includes
the exceptional Lie algebras as the octonionic case of a construction in terms
of a Jordan algebra of hermitian 3x3 matrices (Tits) or various plane and other
geometries (Freudenthal). We present alternative constructions of the magic
square which explain its symmetry, and show explicitly how the use of split
composition algebras leads to analogues of the matrix Lie algebras su(3), sl(3)
and sp(6). We adapt the magic square construction to include analogues of
su(2), sl(2) and sp(4) for all real division algebras.
