We study embeddings of groups of Lie type
H
H
in characteristic
p
p
into exceptional algebraic groups
G
\mathbf {G}
of the same characteristic. We exclude the case where
H
H
is of type
P
S
L
2
\mathrm {PSL}_2
. A subgroup of
G
\mathbf {G}
is Lie primitive if it is not contained in any proper, positive-dimensional subgroup of
G
\mathbf {G}
.
With a few possible exceptions, we prove that there are no Lie primitive subgroups
H
H
in
G
\mathbf {G}
, with the conditions on
H
H
and
G
\mathbf {G}
given above. The exceptions are for
H
H
one of
P
S
L
3
(
3
)
\mathrm {PSL}_3(3)
,
P
S
U
3
(
3
)
\mathrm {PSU}_3(3)
,
P
S
L
3
(
4
)
\mathrm {PSL}_3(4)
,
P
S
U
3
(
4
)
\mathrm {PSU}_3(4)
,
P
S
U
3
(
8
)
\mathrm {PSU}_3(8)
,
P
S
U
4
(
2
)
\mathrm {PSU}_4(2)
,
P
S
p
4
(
2
)
′
\mathrm {PSp}_4(2)’
and
2
B
2
(
8
)
{}^2\!B_2(8)
, and
G
\mathbf {G}
of type
E
8
E_8
. No examples are known of such Lie primitive embeddings.
We prove a slightly stronger result, including stability under automorphisms of
G
\mathbf {G}
. This has the consequence that, with the same exceptions, any almost simple group with socle
H
H
, that is maximal inside an almost simple exceptional group of Lie type
F
4
F_4
,
E
6
E_6
,
2
E
6
{}^2\!E_6
,
E
7
E_7
and
E
8
E_8
, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup inside the algebraic group.
The proof uses a combination of representation-theoretic, algebraic group-theoretic, and computational means.