In recent decades, there has been a growing focus on investigating elementary vortex flows, largely fueled by a renewed interest in coherent structures, which play a fundamental role in many turbulent flows. Identifying critical conditions of these eddies is essential for understanding the flow behavior and the organization of fluid fields. While global stability analysis enables the identification of critical conditions, it does not offer insight into the mechanisms driving these instabilities. An alternative technique, to traditional hydrodynamic instability theory, is the geometric optics approach that relies on the geometric properties of the underlying base flows to establish local instability criteria. The link between these theories has also been explored to identify the conditions that lead to centrifugal, elliptic, and hyperbolic inviscid instabilities. In this study, we discern non-viscous mechanisms through structural sensitivity analysis. The key ingredient lies in making viscosity effects vanish solely in the stability equations while maintaining the base flow field constant, computed at the critical Reynolds number. As a test case, we investigate the flow in a planar sudden expansion. Our findings indicate that classical structural sensitivity accurately identifies the instability core within the recirculation bubble. However, only the inviscid structural sensitivity field clearly reveals that the instability core is concentrated around the centre of an elliptical vortex: an indicator of elliptic instability. To validate our findings, we compare global sensitivity results with asymptotic outcomes.