<p>The full-rank decomposition of reduced biquaternion matrices is fundamental in quaternion algebra and critical for its engineering applications. However, the high computational complexity of biquaternion operations and the lack of unified explicit expressions remain key challenges. We investigate the full-rank decomposition of row or column full-rank reduced biquaternion matrices, derive its explicit expression via their complex representation, and propose a novel numerical algorithm that only involves complex field operations, with theoretical analysis verifying its correctness. We also theoretically demonstrate the decomposition algorithm’s applicability to idempotent matrix factorization and least squares matrix equation solving. Numerical experiments confirm the decomposition algorithm’s efficiency and simplicity, especially in handling the aforementioned matrix tasks. We further apply this decomposition to color image encryption and decryption in image processing and experimental results prove the corresponding encryption–decryption algorithm’s effectiveness and robustness. This research enriches the theoretical system of reduced biquaternion matrix decomposition and provides a practical solution for related engineering scenarios.</p>