On twisted reality conditions

We study the twisted reality condition of Brzeziński et al. (Math Phys Anal Geom 19(3):11, 2016), for spectral triples, in particular with respect to the product and the commutant. Motivated by this, we present the procedure, which allows one to untwist the twisted spectral triples studied in Landi and Martinetti (Lett Math Phys 106:1499–1530, 2016). We also relate this construction to conformally rescaled real twisted spectral triples and discuss the untwisting of the “minimal twist” procedure of an even spectral triple.