Multicomponent Gaussian beam migration methods of elastic waves are currently mainly focused on PP/PS imaging, which is a subset of full elastic-wave imaging. Owing to the employment of pure P-wave explosive sources, these methods only require mode decomposition on the receiver side. However, elastic wave imaging also includes SP and SS cases. SP and SS waves, as part of the complete elastic wavefield, can provide valuable complementary supplementary insights to conventional PP and PS images, because each wave mode offers a distinct illumination pattern, a unique reflectivity behavior, and various amounts and types of information about the geology and rock/fluid properties. To obtain complete reflection elastic-wave images, we propose an extension of the elastic Gaussian beam migration method to multicomponent source and receiver seismic data. The use of multicomponent sources allows for the generation of rich direct-S modes in areas without local interfaces and the ability to send strong S-waves into the deep subsurface. To implement this method, we initially formulate continuation and mode decomposition formulations for free-surface, ocean-bottom, and free-space models. These are derived from the elastic Kirchhoff-Helmholtz integral coupled with corresponding complete boundary conditions on both the source and receiver sides. Such a framework allows for the decomposition of both sides to be automatically performed during migration. With our approach, multicomponent data are decomposed into PP-, PS-, SP-and SS-waves during migration without prior separation, leading to more accurate results. Furthermore, we introduce a vector cross-correlation imaging condition in tensor form to address the polarity reversal issue associated with converted waves. We demonstrate the effectiveness of the proposed method using two synthetic data tests on the free-space model, which corresponds to the case of finite-difference modeling with an absorbing top boundary. It is important to note that multicomponent wavefields are excited separately in a time-series manner using point forces in mutually perpendicular directions.