We consider a rank one group
G
=
⟨
A
,
B
⟩
G = \langle A,B \rangle
acting cubically on a module
V
V
, this means
[
V
,
A
,
A
,
A
]
=
0
[V,A,A,A] =0
but
[
V
,
G
,
G
,
G
]
≠
0
[V,G,G,G] \ne 0
. We have to distinguish whether the group
A
0
:=
C
A
(
[
V
,
A
]
)
∩
C
A
(
V
/
C
V
(
A
)
)
A_0 :=C_A([V,A]) \cap C_A(V/C_V(A))
is trivial or not. We show that if
A
0
A_0
is trivial,
G
G
is a rank one group associated to a quadratic Jordan division algebra. If
A
0
A_0
is not trivial (which is always the case if
A
A
is not abelian), then
A
0
A_0
defines a subgroup
G
0
G_0
of
G
G
acting quadratically on
V
V
. We will call
G
0
G_0
the quadratic kernel of
G
G
. By a result of Timmesfeld we have
G
0
≅
S
L
2
(
J
,
R
)
G_0 \cong \mathrm {SL}_2(J,R)
for a ring
R
R
and a special quadratic Jordan division algebra
J
⊆
R
J \subseteq R
. We show that
J
J
is either a Jordan algebra contained in a commutative field or a Hermitian Jordan algebra. In the second case
G
G
is the special unitary group of a pseudo-quadratic form
π
\pi
of Witt index
1
1
, in the first case
G
G
is the rank one group for a Freudenthal triple system. These results imply that if
(
V
,
G
)
(V,G)
is a quadratic pair such that no two distinct root groups commute and
c
h
a
r
V
≠
2
,
3
\mathrm {char} V\ne 2,3
, then
G
G
is a unitary group or an exceptional algebraic group.