We present high-resolution interferometric imaging of LH 850.02, the brightest 850- and 1200-mu m submillimetre (submm) galaxy in the Lockman Hole. Our observations were made at 890 mu m with the Submillimetre Array (SMA). Our high-resolution submm imaging detects LH 850.02 at greater than or similar to 6 sigma as a single compact (size less than or similar to 1 arcsec or less than or similar to 8 kpc) point source and yields its absolute position to similar to 0.2-arcsec accuracy. LH 850.02 has two alternative radio counterparts within the Submillimetre Common User Bolometer Array (SCUBA) beam (LH 850.02N and LH 850.02S), both of which are statistically very unlikely to be so close to the SCUBA source position by chance. However, the precise astrometry from the SMA shows that the submm emission arises entirely from LH 850.02N, and is not associated with LH 850.02S (by far the brighter of the two alternative identifications at 24 mu m). Fits to the optical-infrared (IR) multicolour photometry of LH 850.02N and LH 850.02S indicate that both lie at z approximate to 3.3, and are therefore likely to be physically associated. At these redshifts, the 24-mu m-to-submm flux density ratios suggest that LH 850.02N has an Arp 220-type starburst-dominated far-IR spectral energy distribution (SED), while LH 850.02S is more similar to Mrk 231, with less dust enshrouded star formation activity, but a significant contribution at 24 mu m (rest frame 5-6 mu m) from an active nucleus. This complex mix of star formation and active galactic nucleus (AGN) activity in multicomponent sources may be common in the high-redshift ultraluminous galaxy population, and highlights the need for precise astrometry from high-resolution interferometric imaging for a more complete understanding.