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We present new ALMA 850µm continuum observations of the original Lyman-α Blob (LAB) in the SSA22 field at z = 3.1 (SSA22-LAB01). The ALMA map resolves the previously identified submillimeter source into three components with obscured star formation rates of ∼200 M⊙ yr−1 . The ALMA sources are associated with several faint rest-frame ultraviolet sources identified in Hubble Space Telescope Imaging Spectrograph (STIS) imaging, at least one of which is spectroscopically confirmed with Keck MOSFIRE to lie within 20 pkpc and 250 km s −1 of one of the ALMA components. We postulate that some of these STIS sources represent a population of low-mass star-forming satellites surrounding the central submillimeter sources, potentially contributing to their growth and activity through accretion. Using a high resolution cosmological zoom simulation of a 1013M⊙ halo at z = 3, including stellar, dust and Lyα radiative transfer, we can accurately model the ALMA+STIS observations and demonstrate that Lyα photons escaping from the central submillimeter sources are expected to resonantly scatter in neutral hydrogen associated with the surrounding satellites: the ALMA sources are effectively ´ılluminatingt ´he halo substructure from within.