Diagonalization of elements of Freudenthal \(R\)-vector space and split Freudenthal \(R\)-vector space. (Q1411740)

This paper deals with two generalizations of the well known fact that any Hermitian matrix \(X \in M(n,K)\), \(K=\mathbb R,\) \(\mathbb C\), or \(\mathbb H\), can be diagonalized by an element of \(SO(n), SU(n)\), or \(Sp(n)\), respectively. The results of this paper finish the diagonalization problem for all the representation spaces appearing in the Freudenthal-Yokota construction of the exceptional Lie groups \(F_4\), \(E_6\), and \(E_7\). In particular, denote by \(\mathfrak J^1\) the Jordan algebra defined as all \(X \in M(3,\mathfrak C)\) satisfying \(I_1 X^* I_1 = X\) where \(\mathfrak C\) is the nonsplit Cayley algebra over \(\mathbb R\) and \(I_1 = diag(-1,1,1)\). The author shows that any element of \(\mathfrak J^1\) can be diagonalized by an element of \(F_4\). The author also shows that any element of the Freudenthal \(\mathbb R\)-vector space \(\mathfrak B\) (respectively split Freudenthal \(\mathbb R\)-vector space \(\mathfrak B'\)) can be diagonalized by an element of \((U(1) \times E_6)/\mathbb Z_3 \subset E_{7(-25)}\) (respectively, \(SU(8)/\mathbb Z_2 \subset E_{7(7)}\)).