of the smallest such matrix yields $$M(\phi)_1=
\left(\begin{array}{cccc}
 0 &   X_1 &      X_2 &     -X_0+X_3 \\
    X_1 & 0   &     -X_0-X_3 & X_2 \\
    -X_2 &  -X_0-X_3 & 0       & X_1  
\end{array}\right).$$ Here is a Macaulay2 [@M2] code computing this matrix (and others by tuning its inputs: the parameterization $\phi$ and the integer $\nu$):

        >A=QQ[s0,s1,s2];
        >f0=s0^2+s1^2+s2^2; f1=2*s0*s2; f2=2*s0*s1; f3=s0^2-s1^2-s2^2; 
        >F=matrix{{f0,f1,f2,f3}};
        >Z1=kernel koszul(1,F);
        >R=A[X0,X1,X2,X3];
        >d=(degree f0)_0; nu=2*(d-1)-1;
        >Z1nu=super basis(nu+d,Z1);
        >Xnu=matrix{{X0,X1,X2,X3}}*substitute(Z1nu,R);
        >Bnu=substitute(basis(nu,A),R); 
        >(m,M)=coefficients(Xnu,Variables=>{s0_R,s1_R,s2_R},Monomials=>Bnu);
        >M -- this is the matrix representation in degree nu

A primary decomposition of the $3\times 3$ minors of $M(\phi)_1$, i.e. ${\mathfrak F}^0_1(\phi)$, returns $$(X_0^2-X_1^2-X_2^2-X_3^2)\cap(X_2,X_1,X_0^2+2X_0X_3+X_3^2)$$ which corresponds to the implicit equation of the sphere plus one embedded double point $(1:0:0:-1)$. Now, a primary decomposition of the $2\times 2$ minors of $M(\phi)_1$, i.e. ${\mathfrak F}^1_1(\phi)$, returns $$(X_2,X_1,X_0+X_3)\cap (X_3,X_2^2,X_1X_2,X_0X_2,X_1^2,X_0X_1,X_0^2)$$ which corresponds to the same embedded point $(1:0:0:-1)$, now with multiplicity one, plus an additional component supported at the origin. Finally, the ideal of $1$-minors of $M(\phi)_1$, i.e. ${\mathfrak F}^2_1(\phi)$, is supported at the origin (i.e. is empty as a subscheme of ${\mathbb P}^2$).

The point $(1:0:0:-1)$ is actually a singular point of the parameterization $\phi$ (but not of the sphere itself). Indeed, the line $L=(0:s_1:s_2)$ is a ${\mathbb P}^1$ that is 