  The origin of intrinsic quantum criticality in the heavy-fermion
superconductor $\beta$-YbAlB$_4$ has been attributed to strong Yb valence
fluctuations and its peculiar crystal structure. Here, we assess these
contributions individually by studying the isostructural but fixed-valence
compound $\beta$-LuAlB$_4$. Quantum oscillation measurements and DFT
calculations reveal a Fermi surface markedly different from that of
$\beta$-YbAlB$_4$, consistent with a `large' Fermi surface there. We also find
an unexpected in-plane anisotropy of the electronic structure, in contrast to
the isotropic Kondo hybridization in $\beta$-YbAlB$_4$.
