  We introduce a new class of $\mathfrak{sl}_2$-triples in a complex simple Lie
algebra $\mathfrak{g}$, which we call magical. Such an $\mathfrak{sl}_2$-triple
canonically defines a real form and various decompositions of $\mathfrak{g}$.
Using this decomposition data, we explicitly parameterize special connected
components of the moduli space of Higgs bundles on a compact Riemann surface
$X$ for an associated real Lie group, hence also of the corresponding character
variety of representations of $\pi_1X$ in the associated real Lie group. This
recovers known components when the real group is split, Hermitian of tube type,
or $\mathrm{SO}_{p,q}$ with $1<p\leq q$, and also constructs previously unknown
components for the quaternionic real forms of $\mathrm{E}_6$, $\mathrm{E}_7$,
$\mathrm{E}_8$ and $\mathrm{F}_4$. The classification of magical
$\mathfrak{sl}_2$-triples is shown to be in bijection with the set of
$\Theta$-positive structures in the sense of Guichard--Wienhard, thus the
mentioned parameterization conjecturally detects all examples of higher
Teichm\"uller spaces. Indeed, we discuss properties of the surface group
representations obtained from these Higgs bundle components and their relation
to $\Theta$-positive Anosov representations, which indicate that this
conjecture holds.
