Autoregressive Initial Bits (ArIB), a framework that combines subimage autoregression and latent variable models, has shown its advantages in lossless image compression. However, in current methods, the image splitting makes the information of latent variables being uniformly distributed in each subimage, and causes inadequate use of latent variables in addition to posterior collapse. To tackle these issues, we introduce Bit Plane Slicing (BPS), splitting images in the bit plane dimension with the considerations on different importance for latent variables. Thus, BPS provides a more effective representation by arranging subimages with decreasing importance for latent variables. To solve the problem of the increased number of dimensions caused by BPS, we further propose a dimension-tailored autoregressive model that tailors autoregression methods for each dimension based on their characteristics, efficiently capturing the dependencies in plane, space, and color dimensions. As shown in the extensive experimental results, our method demonstrates the superior compression performance with comparable inference speed, when compared to the state-of-the-art normalizing-flow-based methods. The code is at https://github.com/ZZ022/ArIB-BPS.