We　study　the　classification　of　entanglement　in　tripartite　systems　by　using　Bell-type　inequalities　and　principal　basis.　By　using　Bell　functions　and　the　generalized　three　dimensional　Pauli　operators,　we　present　a　set　of　Bell　inequalities　which　classifies　the　entanglement　of　triqutrit　fully　separable　and　bi-separable　mixed　states.　By　using　the　correlation　tensors　in　the　principal　basis　representation　of　density　matrices,　we　obtain　separability　criteria　for　fully　separable　and　bi-separable　2　⊗　2　⊗　3　quantum　mixed　states.　Detailed　example　is　given　to　illustrate　our　criteria　in　classifying　the　tripartite　entanglement.　©　2022　The　Author(s).