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.
ηaμν is the 't Hooft symbol:
In other words, they are defined by
where the latter are the anti-self-dual 't Hooft symbols.
More explicitly, these symbols are
They satisfy the self-duality and the anti-self-duality properties:
Some other properties are
The same holds for except for
Obviously due to different
Many properties of these are tabulated in the appendix of 't Hooft's paper and also in the article by Belitsky et al.