P}_{A,\psi,\tilde\psi}$ project the composite bosons and fermions into components with the quantum numbers of the elementary bosons and fermions, with $Y=T^{3R}+T^X$. Once we choose the charges of the composite fermions under \[SU(2)$_L\times$SU(2)$_R\times$U(1)$_X$\]$^{cp}$ the model is fixed. The parameter space spans over the elementary and composite couplings, the composite masses and the fermionic mixings.

The non-trivial mixings preserve the SM gauge symmetry and give rise to massless gauge bosons and fermions (before EWSB): $A_\mu$, $\psi_L$ and $\tilde\psi_R$, defined by: $$\begin{aligned}
\begin{bmatrix}A_\mu \\ A_\mu^*\end{bmatrix}=\begin{bmatrix}\cos\theta_A & \sin\theta_A \\ -\sin\theta_A & \cos\theta_A \end{bmatrix} \begin{bmatrix} A^{el}_\mu \\ {\cal P}_A A^{cp}_\mu \end{bmatrix} \ , 
\qquad \tan\theta_A=\frac{g_{el}}{g_{cp}} \ , \label{ASM}\\
\begin{bmatrix}\psi_L \\ \psi_L^*\end{bmatrix}=\begin{bmatrix}\cos\theta_\psi & \sin\theta_\psi \\ -\sin\theta_\psi & \cos\theta_\psi \end{bmatrix} \begin{bmatrix} \psi_L^{el} \\ {\cal P}_\psi \psi_L^{cp} \end{bmatrix} \ , 
\qquad \tan\theta_\psi=\frac{\Delta_\psi}{m_\psi} \ , \label{psiLSM}\\
\begin{bmatrix}\tilde\psi_R \\ \tilde\psi_R^*\end{bmatrix}=\begin{bmatrix}\cos\theta_{\tilde\psi} & \sin\theta_{\tilde\psi} \\ -\sin\theta_{\tilde\psi} & \cos\theta_{\tilde\psi} \end{bmatrix} \begin{bmatrix} \tilde\psi_R^{el} \\ {\cal P}_{\tilde\psi} \tilde\psi_R^{cp} \end{bmatrix} \ , 
\qquad \tan\theta_{\tilde\psi}=\frac{\tilde\Delta_{\tilde\psi}}{\tilde m_{\tilde\psi}} \ .\label{psiRSM} \end{aligned}$$ The field combinations orthogonal to the massless ones: $A^*_\mu$, $\psi^*$ and $\tilde\psi^*$, have masses: $$\label{M}
M_\phi=\frac{m_\phi}{\cos\theta_\phi}\ , \qquad \phi=A,\psi,\tilde\psi\ .$$ The composite fields that do not mix with the elementary fields (before EWSB) can be written as: $$\tilde{\cal P}_\phi \phi^{cp}\ , \qquad \tilde{\cal P}_\phi\equiv 1-{\cal P}_\phi\ , \qquad 