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This research investigates the disentanglement property within the $\beta$-VAE framework, focusing on unraveling latent representations into semantically meaningful and independent factors. Disentanglement is characterized by variations in a single latent variable corresponding to variations in a specific generative parameter, yielding interpretable and transferable representations. The goal is to achieve a disentangled representation where each latent variable exclusively captures a distinct generative factor, elucidating the underlying factors of variation in the data. The study specifically explores disentanglement in a linear Gaussian setting using the $\gamma\lambda$-VAE model, chosen for its controllability of reconstruction error. We propose a mutual information-based metric, $I_m$, where $m$ denotes the dimension of the latent variable, to assess disentanglement across three scenarios: no correlation, partial correlation, and full correlation among generative variables. Additionally, we conduct a comparative analysis between our metric and contemporary metrics like the SAP metric in the linear Gaussian framework across the aforementioned scenarios. This comparative examination aims to reveal the strengths and limitations of each metric in assessing disentanglement.