Odd magical triples and maximal Higgs bundles

We introduce the notion of extended magical $\mathfrak{sl}_2$-triples, a generalization of the magical $\mathfrak{sl}_2$-triples in Bradlow--Collier--Garc\'ia-Prada--Gothen--Oliveira's work and show that, apart from the known even magical triples, there are precisely three odd triples of nontube type Hermitian Lie algebras that are extended magical. We then show that the Slodowy slice of an odd extended magical triple of $G^\textbf{R}$ in the $G^\textbf{R}$-Higgs bundle moduli space are precisely the maximal components. Finally, assuming that the underlying curve has sufficiently large genus, we give a geometric characterization of extended magical triples and prove a Cayley correspondence for the maximal components of $G^\textbf{R}$-Higgs bundles for nontube type Hermitian Lie groups.