In addressing the challenge of motion artifacts in Positron Emission Tomography (PET) lung scans, our study introduces the Triple Equivariant Motion Transformer (TEMT), an innovative, unsupervised, deep-learningbased framework for efficient respiratory motion correction in PET imaging. Unlike traditional techniques, which segment PET data into bins throughout a respiratory cycle and often face issues such as inefficiency and overemphasis on certain artifacts, TEMT employs Convolutional Neural Networks (CNNs) for effective feature extraction and motion decomposition.TEMT's unique approach involves transforming motion sequences into Lie group domains to highlight fundamental motion patterns, coupled with employing competitive weighting for precise target deformation field generation. Our empirical evaluations confirm TEMT's superior performance in handling diverse PET lung datasets compared to existing image registration networks. Experimental results demonstrate that TEMT achieved Dice indices of 91.40%, 85.41%, 79.78%, and 72.16% on simulated geometric phantom data, lung voxel phantom data, cardiopulmonary voxel phantom data, and clinical data, respectively. To facilitate further research and practical application, the TEMT framework, along with its implementation details and part of the simulation data, is made publicly accessible at https://github.com/yehaowei/temt.