The magnetic anisotropy energy (MAE) in the honeycomb lattice $\mathrm{Cr}{\mathrm{I}}_3$ has attracted much attention due to the fact that it is crucial for the stability of two-dimensional (2D) long-range magnetism. Numerous calculations and experiments demonstrate that $\mathrm{Cr}{\mathrm{I}}_3$ exhibits Kitaev interaction, which is typically linked to edge-sharing octahedral coordination. However, significant lattice distortions in $\mathrm{Cr}{\mathrm{I}}_3$ disrupt the ideal octahedral geometry, leading to additional off-diagonal interactions beyond the conventional Kitaev term. Here we perform a systematic first-principles analysis of the MAE in monolayer $\mathrm{Cr}{\mathrm{I}}_3$ and unambiguously quantify the symmetric off-diagonal exchange ${{\mathrm{\Gamma }}^{\prime }}_1$ in the ferromagnetic (FM) state, a term largely ignored in previous $J\text{−}K\text{−}\mathrm{\Gamma }\text{−}{A}_{\mathbf{k}}$ models. We using a combined energy mapping and angular-dependent MAE approach to determine that $\mathrm{Cr}{\mathrm{I}}_3$ indeed manifests a multineighbor $J\text{−}{K}_1\text{−}{{\mathrm{\Gamma }}^{\prime }}_1\text{−}{K}_2\text{−}{A}_{\mathbf{k}}$ model. Moreover, the ${{\mathrm{\Gamma }}^{\prime }}_1$ contribution in the FM phase exceeds all other anisotropic terms, and meanwhile the single-ion anisotropy $A_{\mathbf{k}}$ largely compensates the remaining bond-dependent contributions. Therefore, ${{\mathrm{\Gamma }}^{\prime }}_1$ not only provides the principal stabilization of 2D FM but also sets the out-of-plane easy axis. Finally, we analyze the strain dependence of interaction parameters and find that lattice expansion amplifies the Kitaev-related terms, highlighting a tunable pathway for spin-orbit coupling enhancement in 2D magnets. Our methodology leverages continuous variation MAE principles to enable precise decomposition of magnetic parameters and rigorously demonstrate the interactions.

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• Antiferroelectricity
• Density of states
• Electrical properties
• Electronic structure
• Magnetic anisotropy
• Magnetic interactions
• Magnetism
• Spin-orbit coupling
• 2-dimensional systems
• Magnetic semiconductors
• Transition metal trichalcogenides
• Density functional theory
• First-principles calculations