In this paper, we introduce the spectral projection operators \(\mathbb {P}_m\) on non-degenerate nilpotent Lie groups \(\mathcal {N}\) of step two, associated to the joint spectrum of sub-Laplacian and derivatives in step two. We construct their kernels \(P_m(\textbf{y},\textbf{t})\) by using Laguerre calculus and find a simple integral representation formula for \(\textbf{y}\ne 0\) . Then we show the kernels are Lipschitzian homogeneous functions on \(\mathcal {N}\setminus \{\textbf{0}\}\) by analytic continuation. Moreover, they are shown to be Calderón–Zygmund kernels, so that the spectral projection operator \(\mathbb {P}_m\) can be extended to a bounded operator from \(L^p(\mathcal {N})\) to itself. We also prove a convergence theorem of the Abel sum \(\lim _{R \rightarrow 1^-} \sum _{m=0}^{\infty } R^{m}\mathbb {P}_{m}\phi =\phi \) by estimating the \(L^p(\mathcal {N})\) -norms of \(\mathbb {P}_m\) . Furthermore, \(\mathbb {P}_m\) are mutually orthogonal projection operators and \(\sum _{m=0}^{\infty } \mathbb {P}_{m}\phi =\phi \) for \(\phi \in L^2(\mathcal {N})\) .
