In an
l
l
-group
G
G
, this paper defines an
l
l
-subgroup
A
A
to be above an
l
l
-subgroup
B
B
(or
B
B
to be below
A
A
) if for every integer
n
n
,
a
∈
A
a \in A
, and
b
∈
B
b \in B
,
n
(
|
a
|
∧
|
b
|
)
⩽
|
a
|
n(|a| \wedge |b|) \leqslant |a|
. It is shown that for every
l
l
-subgroup
A
A
, there exists an
l
l
-subgroup
B
B
maximal below
A
A
which is closed, convex, and, if the
l
l
-group
G
G
is normal-valued, unique, and that for every
l
l
-subgroup
B
B
there exists an
l
l
-subgroup
A
A
maximal above
B
B
which is saturated: if
0
=
x
∧
y
0 = x \wedge y
and
x
+
y
∈
A
x + y \in A
, then
x
∈
A
x \in A
. Given
l
l
-groups
A
A
and
B
B
, it is shown that every lattice ordering of the splitting extension
G
=
A
×
B
G = A \times B
, which extends the lattice orders of
A
A
and
B
B
and makes
A
A
lie above
B
B
, is uniquely determined by a lattice homomorphism
π
\pi
from the lattice of principal convex
l
l
-subgroups of
A
A
into the cardinal summands of
B
B
. This extension is sufficiently general to encompass the cardinal sum of two
l
l
-groups, the lex extension of an
l
l
-group by an
o
o
-group, and the permutation wreath product of two
l
l
-groups. Finally, a characterization is given of those abelian
l
l
-groups
G
G
that split off below: whenever
G
G
is a convex
l
l
-subgroup of an
l
l
-group
H
H
,
H
H
is then a splitting extension of
G
G
by
A
A
for any
l
l
-subgroup
A
A
maximal above
G
G
in
H
H
.