  The 2D layered insulator $\alpha$-RuCl$_3$ is a candidate material for a
quantum spin-liquid state, which may be realized when a magnetic field
suppresses the antiferromagnetic order present at low temperature. Oscillations
in the field dependence of the thermal conductivity, observed for an in-plane
magnetic field $B$ up to a critical field $B^{\star}$, have been attributed to
exotic charge-neutral fermions, viewed as evidence of a quantum spin-liquid
state between the critical field $B_{c}$ $\simeq 7$ T at which the
antiferromagnetic phase ends and $B^{\star}$. Here we report measurements of
the thermal conductivity of $\alpha$-RuCl$_3$ as a function of magnetic field
up to 15 T applied in two distinct in-plane directions: parallel and
perpendicular to the Ru-Ru bond. We find that the number of oscillations
between $B_{c}$ and $B^{\star}$ is the same for the two field directions even
though the field interval between $B_{c}$ and $B^{\star}$ is different. In
other words, the period of the oscillations is controlled by the transition
fields $B_{c}$ and $B^{\star}$. We conclude that these are not true
oscillations -- coming from putative fermions in a spin-liquid state -- but
anomalies associated with a sequence of magnetic transitions.
