In causal inference about two treatments, Conditional Average Treatment Effects (CATEs) play an important role as a quantity representing an individualized causal effect, defined as a difference between the expected outcomes of the two treatments conditioned on covariates. This study assumes two linear regression models between a potential outcome and covariates of the two treatments and defines CATEs as a difference between the linear regression models. Then, we propose a method for consistently estimating CATEs even under high-dimensional and non-sparse parameters. In our study, we demonstrate that desirable theoretical properties, such as consistency, remain attainable even without assuming sparsity explicitly if we assume a weaker assumption called implicit sparsity originating from the definition of CATEs. In this assumption, we suppose that parameters of linear models in potential outcomes can be divided into treatment-specific and common parameters, where the treatment-specific parameters take difference values between each linear regression model, while the common parameters remain identical. Thus, in a difference between two linear regression models, the common parameters disappear, leaving only differences in the treatment-specific parameters. Consequently, the non-zero parameters in CATEs correspond to the differences in the treatment-specific parameters. Leveraging this assumption, we develop a Lasso regression method specialized for CATE estimation and present that the estimator is consistent. Finally, we confirm the soundness of the proposed method by simulation studies.