set of 3×3 self-adjoint matrices over the octonions, again with multiplication is a 27 dimensional, exceptional Jordan algebra (it is exceptional because the octonions are not associative). This was the first example of an Albert algebra. Its automorphism group is the exceptional Lie group F4. Since over the complex numbers this is the only simple exceptional Jordan algebra up to isomorphism, it is often referred to as "the" exceptional Jordan algebra. Over the real numbers there are three isomorphism classes of simple exceptional Jordan algebras. Derivations and structure algebra A derivation of a Jordan algebra A is an endomorphism D of A such that D(xy) = D(x)y+xD(y). The derivations form a Lie algebra der(A). The Jordan identity implies that if x and y are elements of A, then the endomorphism sending z to x(yz)−y(xz) is a derivation. Thus the direct sum of A and der(A) can be made into a Lie algebra, called the structure algebra of A, str(A). A simple example is provided by the Hermitian Jordan algebras H(A,σ). In this case any element x of A with σ(x)=−x defines a derivation. In many important examples, the structure algebra of H(A,σ) is A. Derivation and structure algebras also form part of Tits' construction of the Freudenthal magic square. Formally real Jordan algebras A (possibly nonassociative) algebra over the real numbers is said to be formally real if it satisfies the property that a sum of n squares can only vanish if each one vanishes individually. In 1932, Jordan attempted to axiomatize quantum theory by saying that the algebra of observables of any quantum system should be a formally real algebra that is commutative (xy = yx) and power-associative (the associative law holds for products involving only x, so that powers of any element x are unambiguously defined). He proved that any such algebra is a Jordan algebra. Not every Jordan algebra is formally real, but classified the finite-dimensional formally real Jordan algebras, also called Euclidean Jordan algebras. Every formally real Jordan algebra can be written as a direct sum of so-called simple ones, which are not themselves direct sums in a nontrivial way. In finite dimensions, the simple formally real Jordan algebras come in four infinite families, together with one exceptional case: The Jordan algebra of n×n self-adjoint real matrices, as above. The Jordan algebra of n×n self-adjoint complex matrices, as above. The Jordan algebra of n×n self-adjoint quaternionic