  It has been shown that the L-type bosonic symmetry-protected-trivial (SPT)
phases with pure gauge anomalous boundary can all be realized via non-linear
$\sigma$-models (NL$\sigma$Ms) of the symmetry group $G$ with various
topological terms. Those SPT phases (called the pure SPT phases) can be
classified by group cohomology ${\cal H}^d(G,\mathbb{R}/\mathbb{Z})$. But there
are also SPT phases with mixed gauge-gravity anomalous boundary (which will be
called the mixed SPT phases). Some of the mixed SPT states were also referred
as the beyond-group-cohomology SPT states. In this paper, we show that those
beyond-group-cohomology SPT states are actually within another type of group
cohomology classification. More precisely, we show that both the pure and the
mixed SPT phases can be realized by $G\times SO(\infty)$ NL$\sigma$Ms with
various topological terms. Through the group cohomology ${\cal H}^d[G\times
SO(\infty),\mathbb{R}/\mathbb{Z}]$, we find that the set of our constructed
L-type SPT phases in $d$-dimensional space-time are classified by $
E^d(G)\rtimes \oplus_{k=1}^{d-1} {\cal H}^k(G,\text{iTO}_L^{d-k})\oplus {\cal
H}^d(G,\mathbb{R}/\mathbb{Z}) $ where $G$ may contain time-reversal. Here
$\text{iTO}_L^d$ is the set of the L-type topologically-ordered phases in
$d$-dimensional space-time that have no topological excitations, and one has
$\text{iTO}_L^1=\text{iTO}_L^2=\text{iTO}_L^4=\text{iTO}_L^6=0$,
$\text{iTO}_L^3=\mathbb{Z}$, $\text{iTO}_L^5=\mathbb{Z}_2$,
$\text{iTO}_L^7=2\mathbb{Z}$. Our construction also gives us the topological
invariants that fully characterize the corresponding SPT and iTO phases.
Through several examples, we show how can the universal physical properties of
SPT phases be obtained from those topological invariants.
