The '**t Hooft η symbol** is a symbol which allows one to express the generators of the SU(2) Lie algebra in terms of the generators of Lorentz algebra. The symbol is a blend between the Kronecker delta and the Levi-Civita symbol. It was introduced by Gerard 't Hooft. It is used in the construction of the BPST instanton.

η^{a}_{μν} is the '**t Hooft symbol**:

η
μ
ν
a
=
{
ϵ
a
μ
ν
μ
,
ν
=
1
,
2
,
3
−
δ
a
ν
μ
=
4
δ
a
μ
ν
=
4
0
μ
=
ν
=
4
.
In other words, they are defined by

(
a
=
1
,
2
,
3
;
μ
,
ν
=
1
,
2
,
3
,
4
;
ϵ
1234
=
+
1
)

η
a
μ
ν
=
ϵ
a
μ
ν
4
+
δ
a
μ
δ
ν
4
−
δ
a
ν
δ
μ
4
η
¯
a
μ
ν
=
ϵ
a
μ
ν
4
−
δ
a
μ
δ
ν
4
+
δ
a
ν
δ
μ
4
The (anti)self-duality properties are

η
a
μ
ν
=
1
2
ϵ
μ
ν
ρ
σ
η
a
ρ
σ
,
η
¯
a
μ
ν
=
−
1
2
ϵ
μ
ν
ρ
σ
η
¯
a
ρ
σ

Some other properties are

ϵ
a
b
c
η
b
μ
ν
η
c
ρ
σ
=
δ
μ
ρ
η
a
ν
σ
+
δ
ν
σ
η
a
μ
ρ
−
δ
μ
σ
η
a
ν
ρ
−
δ
ν
ρ
η
a
μ
σ
η
a
μ
ν
η
a
ρ
σ
=
δ
μ
ρ
δ
ν
σ
−
δ
μ
σ
δ
ν
ρ
+
ϵ
μ
ν
ρ
σ
,
η
a
μ
ρ
η
b
μ
σ
=
δ
a
b
δ
ρ
σ
+
ϵ
a
b
c
η
c
ρ
σ
,
ϵ
μ
ν
ρ
θ
η
a
σ
θ
=
δ
σ
μ
η
a
ν
ρ
+
δ
σ
ρ
η
a
μ
ν
−
δ
σ
ν
η
a
μ
ρ
,
η
a
μ
ν
η
a
μ
ν
=
12
,
η
a
μ
ν
η
b
μ
ν
=
4
δ
a
b
,
η
a
μ
ρ
η
a
μ
σ
=
3
δ
ρ
σ
.
The same holds for
η
¯
except for

η
¯
a
μ
ν
η
¯
a
ρ
σ
=
δ
μ
ρ
δ
ν
σ
−
δ
μ
σ
δ
ν
ρ
−
ϵ
μ
ν
ρ
σ
.
and

ϵ
μ
ν
ρ
θ
η
¯
a
σ
θ
=
−
δ
σ
μ
η
¯
a
ν
ρ
−
δ
σ
ρ
η
¯
a
μ
ν
+
δ
σ
ν
η
¯
a
μ
ρ
,
Obviously
η
a
μ
ν
η
¯
b
μ
ν
=
0
due to different duality properties.

Many properties of these are tabulated in the appendix of 't Hooft's paper and also in the article by Belitsky et al.