  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 $P_m(\mathbf{y},\mathbf{t})$ by using Laguerre calculus and find
a simple integral representation formula for $\mathbf{y}\neq 0$. Then we show
the kernels are Lipschitzian homogeneous functions on $\mathcal{N}\setminus
\{\mathbf{0}\}$ by analytic continuation. Moreover, they are shown to be
Calder\'{o}n-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 $\phi\in L^2(\mathcal{N})$.
