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We identified a sample of 58 candidate stars with metallicity [Fe/H] ≲ −0.8 that likely belong to the old bulge spheroid stellar population, and analyse their Na and Al abundances from Apache Point Observatory Galactic Evolution Experiment (APOGEE) spectra. In a previous work, we inspected APOGEE-Stellar Parameter and Chemical Abundance Pipeline abundances of C, N, O, Mg, Al, Ca, Si, and Ce in this sample. Regarding Na lines, one of them appears very strong in about 20 per cent of the sample stars, but it is not confirmed by other Na lines, and can be explained by sky lines, which affect the reduced spectra of stars in a certain radial velocity range. The Na abundances for 15 more reliable cases were taken into account. Al lines in the H band instead appear to be very reliable. Na and Al exhibit a spread in abundances, whereas no spread in N abundances is found, and we found no correlation between them, indicating that these stars could not be identified as second-generation stars that originated in globular clusters. We carry out the study of the behaviour of Na and Al in our sample of bulge stars and literature data by comparing them with chemodynamical evolution model suitable for the Galactic bulge. The Na abundances show a large spread, and the chemodynamical models follow the main data, whereas for aluminum instead, the models reproduce very satisfactorily the nearly secondary-element behaviour of aluminum in the metallicity range below [Fe/H] ≲ −1.0. For the lower-metallicity end ([Fe/H < −2.5), hypernovae are assumed to be the main contributor to yields. [ABSTRACT FROM AUTHOR]