  A construction of the magic square, and hence of exceptional Lie algebras, is
carried out using trialities rather than division algebras. By way of
preparation, a comprehensive discussion of trialities is given, incorporating a
number of novel results and proofs. Many of the techniques are closely related
to, or derived from, ideas which are commonplace in theoretical physics. The
importance of symmetric spaces in the magic square construction is clarified,
allowing the Jacobi property to be verified for each algebra in a uniform and
transparent way, with no detailed calculations required. A variation on the
construction, corresponding to other symmetric spaces, is also given.
