We determine the main properties of the Galactic binary black hole (BBH) population detectable by LISA and strategies to distinguish them from the much more numerous white dwarf binaries. We simulate BBH populations based on cosmological simulations of Milky Way-like galaxies and binary evolution models. We then determine their gravitational wave emission as observed by LISA and build mock catalogs. According to our model LISA will detect $approx4(6)$ binary black holes assuming 4(10) years of operations. Those figures grow to $approx6(9)$ when models are re-normalized to the inferred LIGO/Virgo merger rates. About 40%(70%) of the sources will have a good enough chirp mass measurement to separate them from the much lighter white dwarf and neutron star binaries. Most of the remaining sources should be identifiable by their lack of electromagnetic counterpart within $approx100$ pc. These results are robust with respect to the current uncertainties of the BBH merger rate as measured by LIGO/Virgo as well as the global mass spectrum of the binaries. We determine there is a 94 per cent chance that LISA finds at least one of these systems, which will allow us to pinpoint the conditions where they were formed and possibly find unique electromagnetic signatures.
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