We say that a set \(\Omega \subseteq {\mathbb {R}}^n\) with positive Lebesgue measure is a spectral set if \(L^{2}(\Omega )\) admits an orthogonal basis of exponentials. In this paper, we study the spectrality of self-affine tiles. We prove a class of sets \(T:=T(M,D)\) with positive Lebesgue measure satisfying \(MT=\bigcup _{d\in D}(T+d)\) are spectral sets, where \(M\in M_{2}({\mathbb {Z}})\) is an expanding matrix with \(|\det {(M)}|=4\) and \(D=\{{\textbf{0}},\mathbf {\alpha },\beta ,-(\alpha +\beta )\}\subset {\mathbb {Z}}^2\) is a non-collinear digit set. As an application, we conclude that T is a spectral set if and only if it tiles \({\mathbb {R}}^2\) by translations.
