An oxygen cutting nozzle with/without a small pilot flame has been proposed as a device for combustion incineration and manufacturing of fire-resistant CFRP (Carbon Fiber Reinforced Plastic). Carbon fiber is almost impossible to be burned out by even high temperature burnt gas because of extremely high melting point of carbon fiber and a low oxygen concentration in burnt gases. Our preliminary trial using a propane-oxygen premixed flame, however, has revealed that its unburnt premixed gas region of a high oxygen concentration successfully burns the carbon fiber, not in extremely high temperature burnt gas region. In this study, first, oxygen nozzle of 10 mm x 30 mm exit size, which equipped fuel (propane) pipe of inner diameter of 0.8 mm for a pilot ignition flame, has been constructed. Five kinds of test pieces are used for in-oxygen-combustion experiment to obtain its combustion properties; (i) CFRP slender plate (2 mm wide and 1 mm thickness), (ii)slender plate of CFRP-P (Plastic removed CFRP), "G"FCRs("Graphite" Fine Cylindrical Rods) of diameters of 0.38 mm, 0.55 mm and 0.72 mm. Based on video recording results, it is found that once the test pieces are ignited by the pilot flame, the combustions of all the pieces are maintained in a surrounding oxygen gas flow, resulting in reducing the length of the test pieces. The behaviors of the tip positions are measured as fundamental properties for further analytical investigations of CFRP incineration technologies.