Example 3.
A 3dimensional, non abelian Lie algebra, with
[
g
,
g
]
*
Z
(
g
)
and
dim
[
g
,
g
] = 1
.
Consider:
g
=
{
b
0
0
a
0
0
0
0
c
a, b, c
∈
F
}
spanned by
x
=
0
0
0
1
0
0
0
0
0
, y
=
1
0
0
0
0
0
0
0
0
, z
=
0
0
0
0
0
0
0
0
1
This is a subalgebra of
gl
3
(
F
)
satisfying the properties
[
x, y
] =
x,
[
x, z
] =
0
,
[
y, z
] = 0
. It is also isomorphic to the “direct sum” of the algebra presented
at the first example and the 1dimensional algebra
F
(look at the splitting of
the matrix!).
Back to the classification.
•
Dim
g
=
3
→
dim
[
g
,
g
] = 2
Claim:
[
g
,
g
] is abelian.
Proof.
Let
{
x, y
}
be a basis for [
g
,
g
] and pick
z
∈
g

[
g
,
g
]. Notice that
[
x, y
]
∈
[
g
,
g
] thus [
x, y
] =
ax
+
by
for some
a, b
∈
F
Then define
X
:= [
x,
·
] :
g
→
g
This is a linear map denoted
ad
x
. Notice that
Im
(
X
)
⊆
[
g
,
g
]. Thus
X
(
x
) =
0
, X
(
y
) =
ax
+
by, X
(
z
) =
cx
+
dy
. Clearly
Trace
(
X
) =
tr
(
X
) = 0+
b
+0 =
b
.
Consider the corresponding map
Y
:= [
y,
·
] :
g
→
g
Likewise
tr
(
Y
) =

a
.
On the other hand notice that if
u, v
∈
g
then
[[
u, v
]
,
·
]
=
[[
u,
·
]
,
[
v,
·
]]
=
[
u,
·
]
◦
[
v,
·
]

[
v,
·
]
◦
[
u,
·
]
where
◦
is composition of linear maps. Thus:
tr
[[
u, v
]
,
·
]
=
tr
([
u,
·
]
◦
[
v,
·
]

[
v,
·
]
◦
[
u,
·
])
=
0
2