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We present an analysis of the BVRI photometry of the blazar BL Lacertae on diverse timescales from 2020 mid-July to mid-September. We have used 11 different optical telescopes around the world and have collected data over 84 observational nights. The observations cover the onset of a new activity phase of BL Lacertae that started in 2020 August (termed as the 2020 August flare by us), and the analysis is focused on the intranight variability. On short-term timescales, (i) flux varied with ∼2.2 mag in the R band, (ii) the spectral index was found to be weakly dependent on the flux (i.e., the variations could be considered mildly chromatic), and (iii) no periodicity was detected. On intranight timescales, BL Lacertae was found to predominantly show bluer-when-brighter chromatism. We also found two cases of significant interband time lags of the order of a few minutes. The duty cycle of the blazar during the 2020 August flare was estimated to be quite high (∼90% or higher). We decomposed the intranight light curves into individual flares and determined their characteristics. On the basis of our analysis and assuming the turbulent jet model, we determined some characteristics of the emitting regions: Doppler factor, magnetic field strength, electron Lorentz factor, and radius. The radii determined were discussed in the framework of the Kolmogorov theory of turbulence. We also estimated the weighted mean structure function slope on intranight timescales, related it to the slope of the power spectral density, and discussed it with regard to the origin of intranight variability.