  Quartic spectrahedra in 3-space form a semialgebraic set of dimension 24.
This set is stratified by the location of its ten nodes. There are twenty
maximal strata, identified recently by Degtyarev and Itenberg, via the global
Torelli Theorem for real K3 surfaces. We here give a new proof that is
self-contained and algorithmic. This involves extending Cayley's
characterization of quartic symmetroids, by the property that the branch locus
of the projection from a node consists of two cubic curves. This paper
represents a first step towards the classification of all spectrahedra of a
given degree and dimension.
