An interval-valued fuzziness structure is an effective approach addressing ambiguity and for expressing people's hesitation in everyday situations. An $\mathcal{N}$-structure is a novel technique for solving practical problems. This is beneficial for resolving a variety of issues, and a lot of progress is being made right now. In order to develop crossing cubic structures ( $\mathcal{C}\mathcal{C}\mathcal{S}s$), Jun et al. amalgamate interval-valued fuzziness and $\mathcal{N}$-structures. In this manuscript, our main contribution is to originate the concepts of crossing cubic ( $\mathcal{C}\mathcal{C}$) Lie algebra, $\mathcal{C}\mathcal{C}$ Lie sub-algebra, ideal, and homomorphism. We investigate some properties of these concepts. In a Lie algebra, the construction of a quotient Lie algebra via the $\mathcal{C}\mathcal{C}$ Lie ideal is provided. Furthermore, the $\mathcal{C}\mathcal{C}$ isomorphism theorems are presented.