$v\in
V$, the elements of $K$ turn out to be block diagonal matrices leaving invariant the following subspaces $V^{A}$, $1 \leq A \leq
3$: $$\begin{aligned}
V^{1}& = &\{(x_1,0,0,0,0) \in V | x_1 \in \real\}\,, \nonumber \\
V^{2}& = &\{(0,x_2,x_5,0,0) \in V | x_2\,,x_5  \in \real\} \,,\nonumber \\
V^{3}& = &\{(0,0,0,x_7,x_8) \in V | x_7\,,x_8 \in \real\}\,.\end{aligned}$$

A proper rotation $r(\phi)\in$ O$_1(2)$ of an angle $\phi$ and the reflection diag(-1,1)$\in {\rm O}_1(2)$ are represented in $V$ by the $5\times 5$ matrices $1\oplus r(-2\phi)\oplus r(\phi)$ and ${\rm diag}(1,-1,1,-1,1)$.

Noting that the complex variables $z_1=x_2+\rmi x_5$ and $z_2=x_7+\rmi x_8$ transform into $\exp(-2\rmi \phi)\,z_1$ and, respectively, $\exp(\rmi \phi)\,z_2$ under a rotation and into $z_1^*$ and, respectively, $-z_2^*$ under a reflection, it is easy to realize that a possible MIB for $(K,
V)$ is the following: $$\begin{aligned}
\lambda_1 &=& x_1 \,,\nonumber\\
\lambda_2 &=&|z_1|^2\,=\, x_2^2 + x_5^2\,,\nonumber\\
\lambda_3 &=& |z_2|^2\,=\,x_7^2 + x_8^2\,,\nonumber\\
\lambda_4 &=& 2 \, \mbox{\rm Re} (z_1 {z_2}^2)\,=\,2
\left( x_2 {x_7}^2 -2 x_5 x_7 x_8 - x_2 {x_8}^2 \right)\;.\end{aligned}$$

It is, now, easy to express the $p$’s in term of the $\lambda$’s, $p=\phi(\lambda)$, and to check that the following expressions, obtained in this way, identically satisfy the equation $A(p(\lambda))=0$ (see (\[attivo\])) entering the definition 