Automatic Discovery of One-Parameter Subgroups of Lie Groups: Compact and Non-Compact Cases of SO(n) and SL(n)

We introduce a novel framework for the automatic discovery of one-parameter subgroups (H_γ) of $SO(3)$ and, more generally, $SO(n)$. One-parameter subgroups of $SO(n)$ are crucial in a wide range of applications, including robotics, quantum mechanics, and molecular structure analysis. Our method utilizes the standard Jordan form of skew-symmetric matrices, which define the Lie algebra of $SO(n)$, to establish a canonical form for orbits under the action of H_γ. This canonical form is then employed to derive a standardized representation for H_γ-invariant functions. By learning the appropriate parameters, the framework uncovers the underlying one-parameter subgroup H_γ. The effectiveness of the proposed approach is demonstrated through tasks such as double pendulum modeling, moment of inertia prediction, top quark tagging and invariant polynomial regression, where it successfully recovers meaningful subgroup structure and produces interpretable, symmetry-aware representations.