In 2021, Tang et al. proposed an improved construction of Reed-Solomon (RS) erasure codes with four parity symbols to accelerate the computation of Reed-Muller (RM) transform-based RS algorithm. The idea is to change the original Vandermonde parity-check matrix into a systematic Vandermonde parity-check matrix. However, the construction relies on a computer search and requires that the size of the information vector of RS codes does not exceed $52$. This paper improves its idea and proposes a purely algebraic construction. The proposed method has a more explicit construction, a wider range of codeword lengths, and competitive encoding/erasure decoding computational complexity.