 Conjugate fundamental spinor of SU(2)

For the defining SU(2) action ${\eta }^{\prime }=A\eta$, the conjugate row ${\bar{\eta }}_{\alpha }=({\eta }^{\alpha }{)}^{\ast }$ transforms by $A^{†}=A^{-1}$. Thus ${\bar{\eta }}_{\alpha }{\eta }^{\alpha }$ is invariant. The alternating invariant tensor ${ϵ}_{12}=1$ satisfies $A^TϵA=ϵ$. Choose its inverse ${ϵ}^{12}=-1$; then ${ϵ}^{\alpha \gamma }{ϵ}_{\gamma \beta }={\delta }^{\alpha }{}_{\beta }$. Raising the conjugate index, ${\tilde{\eta }}^{\alpha }={ϵ}^{\alpha \beta }{\bar{\eta }}_{\beta }$, gives another defining spinor. The resulting antilinear intertwiner squares to $-I$, so the defining representation is a pseudoreal representation.