  The electronic nematic phase of Sr$_3$Ru$_2$O$_7$ is investigated by
high-resolution in-plane thermal expansion measurements in magnetic fields
close to 8 T applied at various angles $\Theta$ off the c-axis. At
$\Theta<10^\circ$ we observe a very small ($10^{-7}$) lattice distortion which
breaks the four-fold in-plane symmetry, resulting in nematic domains with
interchanged $a$- and b-axis. At $\Theta \gtrsim 10^\circ$ the domains are
almost fully aligned and thermal expansion indicates an area-preserving lattice
distortion of order $2\times 10^{-6}$ which is likely related to orbital
ordering. Since the system is located in the immediate vicinity to a
metamagnetic quantum critical end point, the results represent the first
observation of a structural relaxation driven by quantum criticality.
