Training Large Language Models (LLMs) presents a significant communication bottleneck, predominantly due to the growing scale of the gradient to communicate across multi-device clusters. However, how to mitigate communication overhead in practice remains a formidable challenge due to the weakness of the methodology of the existing compression methods, especially the neglect of the characteristics of the gradient. In this paper, we consider and demonstrate the low-rank properties of gradient and Hessian observed in LLMs training dynamic, and take advantage of such natural properties to design SEPARATE, a simple low-rank projection for gradient compression in modern large-scale model training processes. SEPARATE realizes dimensional reduction by common random Gaussian variables and an improved moving average error-feedback technique. We theoretically demonstrate that SEPARATE-based optimizers maintain the original convergence rate for SGD and Adam-Type optimizers for general non-convex objectives. Experimental results show that SEPARATE accelerates training speed by up to 2× for GPT-2-Medium pre-training, and improves performance on various benchmarks for LLAMA2-7B fine-tuning.