  Self-testing is a powerful certification of quantum systems relying on
measured, classical statistics. This paper considers self-testing in bipartite
Bell scenarios with small number of inputs and outputs, but with quantum states
and measurements of arbitrarily large dimension. The contributions are twofold.
Firstly, it is shown that every maximally entangled state can be self-tested
with four binary measurements per party. This result extends the earlier work
of Man\v{c}inska-Prakash-Schafhauser (2021), which applies to maximally
entangled states of odd dimensions only. Secondly, it is shown that every
single binary projective measurement can be self-tested with five binary
measurements per party. A similar statement holds for self-testing of
projective measurements with more than two outputs. These results are enabled
by the representation theory of quadruples of projections that add to a scalar
multiple of the identity. Structure of irreducible representations, analysis of
their spectral features and post-hoc self-testing are the primary methods for
constructing the new self-tests with small number of inputs and outputs.
