The primary purpose of this paper is to clarify the relation between previous results in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145], [Amer. J. Math. 141 (2019), pp. 661–703], and [Camb. J. Math. 8 (2020), p. 775–951] via the construction of some interesting locally analytic representations. Let
E
E
be a sufficiently large finite extension of
Q
p
\mathbf {Q}_p
and
ρ
p
\rho _p
be a
p
p
-adic semi-stable representation
G
a
l
(
Q
p
¯
/
Q
p
)
→
G
L
3
(
E
)
\mathrm {Gal}(\overline {\mathbf {Q}_p}/\mathbf {Q}_p)\rightarrow \mathrm {GL}_3(E)
such that the associated Weil–Deligne representation
W
D
(
ρ
p
)
\mathrm {WD}(\rho _p)
has rank two monodromy and the associated Hodge filtration is non-critical. A computation of extensions of rank one
(
φ
,
Γ
)
(\varphi , \Gamma )
-modules shows that the Hodge filtration of
ρ
p
\rho _p
depends on three invariants in
E
E
. We construct a family of locally analytic representations
Σ
m
i
n
(
λ
,
L
1
,
L
2
,
L
3
)
\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)
of
G
L
3
(
Q
p
)
\mathrm {GL}_3(\mathbf {Q}_p)
depending on three invariants
L
1
,
L
2
,
L
3
∈
E
\mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3 \in E
, such that each representation in the family contains the locally algebraic representation
A
l
g
⊗
S
t
e
i
n
b
e
r
g
\mathrm {Alg}\otimes \mathrm {Steinberg}
determined by
W
D
(
ρ
p
)
\mathrm {WD}(\rho _p)
(via classical local Langlands correspondence for
G
L
3
(
Q
p
)
\mathrm {GL}_3(\mathbf {Q}_p)
) and the Hodge–Tate weights of
ρ
p
\rho _p
. When
ρ
p
\rho _p
comes from an automorphic representation
π
\pi
of a unitary group over
Q
\mathbf {Q}
which is compact at infinity, we show (under some technical assumption) that there is a unique locally analytic representation in the above family that occurs as a subrepresentation of the Hecke eigenspace (associated with
π
\pi
) in the completed cohomology. We note that [Amer. J. Math. 141 (2019), pp. 611–703] constructs a family of locally analytic representations depending on four invariants ( cf. (4) in that publication ) and proves that there is a unique representation in this family that embeds into the Hecke eigenspace above. We prove that if a representation
Π
\Pi
in Breuil’s family embeds into the Hecke eigenspace above, the embedding of
Π
\Pi
extends uniquely to an embedding of a
Σ
m
i
n
(
λ
,
L
1
,
L
2
,
L
3
)
\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)
into the Hecke eigenspace, for certain
L
1
,
L
2
,
L
3
∈
E
\mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3\in E
uniquely determined by
Π
\Pi
. This gives a purely representation theoretical necessary condition for
Π
\Pi
to embed into completed cohomology. Moreover, certain natural subquotients of
Σ
m
i
n
(
λ
,
L
1
,
L
2
,
L
3
)
\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)
give an explicit complex of locally analytic representations that realizes the derived object
Σ
(
λ
,
L
_
)
\Sigma (\lambda , \underline {\mathscr {L}})
in (1.14) of [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145]. Consequently, the locally analytic representation
Σ
m
i
n
(
λ
,
L
1
,
L
2
,
L
3
)
\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)
gives a relation between the higher
L
\mathscr {L}
-invariants studied in [Amer. J. Math. 141 (2019), pp. 611–703] as well as the work of Breuil and Ding and the
p
p
-adic dilogarithm function which appears in the construction of
Σ
(
λ
,
L
_
)
\Sigma (\lambda , \underline {\mathscr {L}})
in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145].