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 Coriolis 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.
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