DTA measurements with mixtures of aluminum oxide and lutetium oxide around the 1:1 perovskite composition were performed up to 1970 deg. C. A peak with onset 1901 deg C was due to the melting of the eutectic Lu4Al2O9 (monoclinic phase) and LuAlO3 (perovskite). Neither peritectic melting of the perovskite nor its decomposition in the solid phase could be resolved experimentally. The maximum of the eutectic peak size is near x=0.44, on the Lu-rich side of the perovskite, which is consistent with the conclusion that LuAlO3 melts peritectically at ca. 1907 deg. C as proposed by Wu, Pelton, J. Alloys Compd. 179 (1992) 259. Thermodynamic equilibrium calculations reveal, that under strongly reducing conditions (oxygen partial pressure <10^{-13} bar) aluminum(III) oxide can be reduced to suboxides or even Al metal. It is shown that under such conditions a new phase field with liquid Al can appear. Comment: 6 pages, 4 figures, 1 table, submitted to J. Crystal Growth