't Hooft symbol (${\eta }_{\mu \nu }^a,{\bar{\eta }}_{\mu \nu }^a$)

For the orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d{x}^4$, define ${\eta }_{ij}^a={ϵ}_{aij}$, ${\eta }_{i4}^a={\delta }_{ai}$, and extend antisymmetrically. The three forms $\frac{1}{2}{\eta }_{\mu \nu }^{a}d{x}^{\mu }\wedge d{x}^{\nu }$ are self-dual. Reversing the sign of the components involving index four gives the anti-self-dual symbols $\bar{\eta }$. They identify the two three-dimensional eigenspaces of the Hodge star on differential two-forms and give compact formulas for the BPST instanton.