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

Levi-Civita symbol

The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...

. It was introduced by Gerard 't Hooft. It is used in the construction of the BPST instanton

BPST instanton

The BPST instanton is the instanton with winding number 1 found by Alexander Belavin, Alexander Polyakov, Albert Schwarz and Yu. S. Tyupkin. It is a classical solution to the equations of motion of SU Yang-Mills theory in Euclidean space-time , meaning it describes a transition between two...

.
They are defined by

()



The (anti)self-duality properties are

Some other properties are


The same holds for except for


Obviously 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.