Consider the nest algebra \(\text {Alg}(\mathcal {N})\) relating to a nest \(\mathcal {N}\) based on a Hilbert space with a fixed \(n \ge 2\) in terms of an integer. An \(n\) -linear map that is symmetric and from \(G : \text {Alg}(\mathcal {N})^n \rightarrow \text {Alg}(\mathcal {N})\) is known as a symmetric generalized \(n\) -derivation if there is an associated symmetric \(n\) -derivation \(D : \text {Alg}(\mathcal {N})^n \rightarrow \text {Alg}(\mathcal {N})\) satisfying the condition: \(G(T_1, T_2, \ldots , T_iT'_i, \ldots , T_n)= G(T_1, T_2, \ldots , T_i, \ldots , T_n)T'_i + T_iD(T_1, T_2, \ldots , T'_i, \ldots , T_n),\) for all \(T_1, T_2, \ldots , T_n, T'_i \in \text {Alg}(\mathcal {N}.)\) Structure of nest algebras involving symmetric generalized \(n\) -derivations, focusing on their interaction with square-closed Lie ideals are explored. Conditions are identified under which these ideals are included in the center of the algebra. Moreover, the analysis outlines the trace forms of symmetric generalized \(n\) -derivations that meet specific functional identities. These findings generalize results from \(C^*\) -algebras to nest algebras, enhancing the understanding of derivational frameworks within non-self-adjoint operator algebras.
