Lecture 22
Theorem 1.
(Engel) If each element of
g
is adnilpotent, then
g
is nilpotent.
Proof of a particular case
: Suppose each
x
∈
g
satisfies
ad
2
x
= 0.
That is,
for all
x, y
∈
g
,
[
x,
[
x, y
]] = 0. Claim
g
2
= 0, provided
char
F
6
= 3. That is,
[
x,
[
y, z
]] = 0 for all
x, y, z
∈
g
.
Given
w, z
∈
g
=
⇒
[
z,
[
z, w
]] = 0. Replace
z
by
x
+
y
.
0
=
[
x
+
y,
[
x
+
y, w
]]
=
[
x,
[
x, w
]] + [
x,
[
y, w
]] + [
y,
[
x, w
]] + [
y,
[
y, w
]]
=
0 + [
x,
[
y, w
]] + [
y,
[
x, w
]] + 0
∴
[
x,
[
y, w
]] is skew symmetric in
x, y
Now by Jacobi,
0
=
[
w,
[
x, y
]] + [
x,
[
y, w
]] + [
y,
[
w, x
]]
=
[
w,
[
x, y
]]

[
x,
[
w, y
]]

[
w,
[
y, x
]]
=
[
w,
[
x, y
]] + [
w,
[
x, y
]] + [
w,
[
x, y
]]
=
3[
w,
[
x, y
]]
∴
g
2
= 0 provided
char
F
6
= 3.
Remark
: In this case it turns out that
g
3
= 0.
To prove Engel, we use
Theorem 2.
Let
V
be a finite dimensional vector space and
g
⊂
gl
(
V
)
a
subalgebra. If each element of
g
is nilpotent as a linear map
V
→
V
, then
T
z
∈
g
Ker
(
z
)
6
= 0
.
1