Let
p
≥
7
p \geq 7
be a prime number. Let
S
(
3
)
S(3)
denote the third Morava stabilizer algebra. In recent years, Kato-Shimomura and Gu-Wang-Wu found several families of nontrivial products in the stable homotopy ring of spheres
π
∗
(
S
)
\pi _* (S)
using
H
∗
,
∗
(
S
(
3
)
)
H^{*,*} (S(3))
. In this paper, we determine all nontrivial products in
π
∗
(
S
)
\pi _* (S)
of the Greek letter family elements
α
s
,
β
s
,
γ
s
\alpha _s, \beta _s, \gamma _s
and Cohen’s elements
ζ
n
\zeta _n
which are detectable by
H
∗
,
∗
(
S
(
3
)
)
H^{*,*} (S(3))
. In particular, we show
β
1
γ
s
ζ
n
≠
0
∈
π
∗
(
S
)
\beta _1 \gamma _s \zeta _n \neq 0 \in \pi _*(S)
, if
n
≡
2
n \equiv 2
mod 3,
s
≢
0
,
±
1
s \not \equiv 0, \pm 1
mod
p
p
.