In this paper, we extend the split quaternions to n-split quaternions with \(n\ge 1\) and study matrix equation \(X-A\widehat{X}B=C\) over n-split quaternions based on the semi-tensor product of matrices, where \(\widehat{X}\in \{X, X^{\eta }\}\) , and \(X^{\eta }\) is the \(\eta \) -conjugate of X, \(\eta \in \{\textbf{i}, \textbf{j}, \textbf{k}\}\) . First, by utilizing the semi-tensor product of matrices, we propose the left and right real element representation of split quaternion and study their properties. Then, we provide the real columns stacking form of n-split quaternions and n-split quaternion matrices. Subsequently, we used the above method to convert the n-split quaternions matrix equation \(X-A\widehat{X}B=C\) into a system of real linear equations and found its general solution, and obtain the bisymmetric solution of the matrix equation \(X-A\widehat{X}B=C\) by combining it with the \(\mathcal {H}\) -representation. And then, we give the corresponding algorithm and confirm the effectiveness of the method through numerical experiments. Finally, we combine this method with group permutation and apply it to image encryption and decryption, achieving good results.
