  We systematically study how properties of abstract operator systems help
classifying linear matrix inequality definitions of sets. Our main focus is on
polyhedral cones, the 3-dimensional Lorentz cone, where we can completely
describe all defining linear matrix inequalities, and on the cone of positive
semidefinite matrices. Here we use results on isometries between matrix
algebras to describe linear matrix inequality definitions of relatively small
size. We conversely use the theory of operator systems to characterize special
such isometries.
