The splitting processes in identified quark and gluon jets are investigated using longitudinal and transverse observables. The jets are selected from symmetric three-jet events measured in Z decays with the Delphi detector in 1991-1994. Gluon jets are identified using heavy quark anti-tagging. Scaling violations in identified gluon jets are observed for the first time. The scale energy dependence of the gluon fragmentation function is found to be about two times larger than for the corresponding quark jets, consistent with the QCD expectation TeX . The primary splitting of gluons and quarks into subjets agrees with fragmentation models and, for specific regions of the jet resolution TeX , with NLLA calculations. The maximum of the ratio of the primary subjet splittings in quark and gluon jets is TeX . Due to non-perturbative effects, the data are below the expectation at small TeX . The transition from the perturbative to the non-perturbative domain appears at smaller TeX for quark jets than for gluon jets. Combined with the observed behaviour of the higher rank splittings, this explains the relatively small multiplicity ratio between gluon and quark jets.