In order to discuss the remaining four exceptional Lie groups, we introduce the algebra of $3\times3$ octonionic Hermitian matrices, known as the Albert algebra, written as $\bH_3(\OO)$. Since the product of Hermitian matrices is not necessarily Hermitian, we introduce the Jordan product \begin{equation} \AAA \circ \BBB = {1\over2} (\AAA\BBB + \BBB\AAA) \end{equation} for elements $\AAA,\BBB\in\bH_3(\OO)$, which we henceforth refer to as Jordan matrices. It is easily checked that the Jordan product takes Hermitian matrices to Hermitian matrices, is (obviously) commutative, but is not associative. A Jordan algebra is an algebra with a commutative product $\circ$ that also satisfies \begin{equation} (\AAA\circ\BBB)\circ\AAA^2 = \AAA\circ(\BBB\circ\AAA^2) \end{equation} where of course \begin{equation} \AAA^2 = \AAA\circ\AAA \end{equation} Jordan algebras were introduced by the German mathematical physicist Pascual Jordan in order to represent observables in quantum mechanics. All Jordan algebras except the Albert algebra arise by introducing the Jordan product on an associative algebra; the Albert algebra is therefore also referred to as the exceptional Jordan algebra. Higher powers must be explicitly defined, such as \begin{equation} \AAA^3 = \AAA^2 \circ \AAA = \AAA \circ \AAA^2 \end{equation}

We also introduce the Freudenthal product of two Jordan matrices $A$ and $B$, given by \begin{align} \AAA*\BBB = \AAA \circ \BBB &- \frac12 \Big(\AAA\,\tr(\BBB)+\BBB\,\tr(\AAA)\Big) \\ &+ \frac12 \Big(\tr(\AAA)\,\tr(\BBB)-\tr(\AAA\circ \BBB)\Big) \end{align} where the identity matrix is implicit in the last term. The determinant of a Jordan matrix can then be defined as \begin{equation} \det(\AAA) = \frac13 \, \tr \Big( (\AAA*\AAA) \circ \AAA \Big) \label{Det} \end{equation} Concretely, if \begin{equation} \AAA = \begin{pmatrix} p& \bar{a}& c\\ a& m& \bar{b}\\ \bar{c}& b& n\\ \end{pmatrix} \end{equation} with $p,m,n\in\RR$ and $a,b,c\in\OO$ then \begin{equation} \det\AAA = pmn + c(ba) + \bar{c(ba)} - n|a|^2 - p|b|^2 - m|c|^2 \end{equation}

These products become somewhat less mysterious if we consider the restriction of $\OO$ to $\RR$, and suppose that \begin{equation} \AAA = vv^\dagger, \qquad \BBB = ww^\dagger \end{equation} where $v,w\in\RR^3$ (and of course Hermitian conjugation, denoted by $\dagger$, reduces to the matrix transpose). In this case, \begin{align} \tr(\AAA\circ\BBB) &= (v\cdot w)^2 \\ \AAA*\BBB &= (v\times w)(v\times w)^\dagger \end{align} where $\cdot$ and $\times$ denote the ordinary dot and cross product. The Jordan product $\circ$ and the Freudenthal product $*$ can therefore be thought of as generalizations of the dot and cross products, respectively.