| Summary: | In this paper, the notion of cubic intuitionistic <i>q</i>-ideals in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>C</mi> <mi>I</mi> </mrow> </semantics> </math> </inline-formula>-algebras is introduced. A relationship between a cubic intuitionistic subalgebra, a cubic intuitionistic ideal, and a cubic intuitionistic <i>q</i>-ideal is discussed. Conditions for a cubic intuitionistic ideal to be a cubic intuitionistic <i>q</i>-ideal are provided. Characterizations of a cubic intuitionistic <i>q</i>-ideal are considered. The cubic intuitionistic extension property for a cubic intuitionistic <i>q</i>-ideal is established. Furthermore, the product of cubic intuitionistic subalgebras, ideals, and <i>q</i>-ideals are investigated.
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