  We study the semiclassical limit of a class of invariant tensors for
infinite-dimensional unitary representations of $\mathrm{SL}(2,\mathbb{C})$ of
the principal series, corresponding to generalized Clebsch-Gordan coefficients
with $n\geq3$ legs. We find critical configurations of the quantum labels with
a power-law decay of the invariants. They describe 3d polygons that can be
deformed into one another via a Lorentz transformation. This is defined viewing
the edge vectors of the polygons are the electric part of bivectors satisfying
a (frame-dependent) relation between their electric and magnetic parts known as
$\gamma$-simplicity in the loop quantum gravity literature. The frame depends
on the SU(2) spin labelling the basis elements of the invariants. We compute a
saddle point approximation using the critical points and provide a
leading-order approximation of the invariants. The power-law is universal if
the SU(2) spins have their lowest value, and $n$-dependent otherwise. As a side
result, we provide a compact formula for $\gamma$-simplicity in arbitrary
frames. The results have applications to the current EPRL model, but also to
future research aiming at going beyond the use of fixed time gauge in spin foam
models.
