  This paper reports the first observation of the Jacobi shape transition in
$^{31}$P using high energy $\gamma$-rays from the decay of giant dipole
resonance (GDR) as a probe. The measured GDR spectrum in the decay of $^{31}$P
shows a distinct low energy component around 10 MeV, which is a clear signature
of Corioli's splitting in a highly deformed rotating nucleus.
  Interestingly, a self-conjugate $\alpha$-cluster nucleus $^{28}$Si, populated
at similar initial excitation energy and angular momentum, exhibits a vastly
different GDR line shape. Even though the angular momentum of the compound
nucleus $^{28}$Si is higher than the critical angular momentum required for the
Jacobi shape transition, the GDR lineshape is akin to a prolate deformed
nucleus. Considering the present results for $^{28}$Si and similar observation
recently reported in $^{32}$S, it is proposed that the nuclear orbiting
phenomenon exhibited by $\alpha$-cluster nuclei hinders the Jacobi shape
transition. The present experimental results suggest a possibility to
investigate the nuclear orbiting phenomenon using high energy $\gamma$-rays as
a probe.
