Miehe's spectral decomposition model [1] is widely used in phase-field fracture simulations to correctly capture crack propagation under tensile states while preventing crack growth under compression. However, this approach presents some significant limitations, such as its high computational cost, since for a general three-dimensional stress state an eigenvalue problem must be solved at each Gauss point and at every iteration of the nonlinear solver. Additionally, numerical instabilities may arise when two or more principal strains coincide. To circumvent this issue, numerical perturbations are often introduced, which may adversely affect the accuracy of the results. Moreover, the numerical implementation of the spectral decomposition can be nontrivial, as it has been predominantly addressed in the literature through specific algorithms, such as those presented in [2, 3]. Last but not least, the extension of the spectral decomposition to anisotropic materials is not straightforward and may require distinct formulations, for which the spectral decomposition of the strain tensor is not always adopted [4, 5]. In this context, the present work aims to address these limitations by proposing a systematic basis-independent formulation for the spectral decomposition of the strain tensor. For this, the use of the well-known representation theorems [6] is explored to derive a formulation that does not require the computation of eigenvalues and eigenvectors. Hence, the formulation employs tensorial bases and scalar invariants to express the tensile and compressive parts of the strain tensor, making it independent of any specific coordinate basis. This feature facilitates its extension to anisotropic materials, as the tensorial bases can be constructed to reflect the underlying material symmetry. Numerical examples for isotropic and transversely isotropic cases indicate improved robustness of the proposed approach in scenarios where principal strains coincide, without resorting to numerical perturbations, while maintaining accuracy comparable to Miehe's original formulation.