The conditions  of β-algebra is enforced into the structure of cubic intuitionistic fuzzy settings. Furthermore, the concept of cubic intuitionistic  β-subalgebra  is expressed and its pertinent properties were explored. Also, discussed about the level set of cubic intuitionistic  β-subalgebras and  furnished some fascinating results on the cartesian  product of cubic intuitionistic  β-subalgebra.  Moreover, the notion of -normed cubic intuitionistic  β-subalgebras have been introduced and relevant results are studied.