Don't use paper, use silicon. The matrix $B$ (for the basis $x_{1,1},x_{1,2},x_{1,3},x_{2,2},x_{2,3},x_{3,3}$) is
$$\pmatrix{ 1-{a_{{1,1}}}^{2}&-2\,a_{{1,2}}a_{{1,1}
}&-2\,a_{{1,3}}a_{{1,1}}&-{a_{{1,2}}}^{2}&-2\,a_{{1,3}}a_{{1,2}}&-{a_{
{1,3}}}^{2}\\ -a_{{1,1}}a_{{2,1}}&1-a_{{1,2}}a_{{2,1
}}-a_{{1,1}}a_{{2,2}}&-a_{{1,3}}a_{{2,1}}-a_{{1,1}}a_{{2,3}}&-a_{{1,2}
}a_{{2,2}}&-a_{{1,3}}a_{{2,2}}-a_{{1,2}}a_{{2,3}}&-a_{{1,3}}a_{{2,3}}
\\ -{a_{{2,1}}}^{2}&-2\,a_{{2,2}}a_{{2,1}}&-2\,a_{{2
,3}}a_{{2,1}}&1-{a_{{2,2}}}^{2}&-2\,a_{{2,3}}a_{{2,2}}&-{a_{{2,3}}}^{2
}\\-a_{{1,1}}a_{{3,1}}&-a_{{1,2}}a_{{3,1}}-a_{{1,1}
}a_{{3,2}}&-a_{{1,3}}a_{{3,1}}+1-a_{{1,1}}a_{{3,3}}&-a_{{1,2}}a_{{3,2}
}&-a_{{1,2}}a_{{3,3}}-a_{{1,3}}a_{{3,2}}&-a_{{1,3}}a_{{3,3}}
\\ -a_{{2,1}}a_{{3,1}}&-a_{{2,2}}a_{{3,1}}-a_{{2,1}}
a_{{3,2}}&-a_{{2,3}}a_{{3,1}}-a_{{2,1}}a_{{3,3}}&-a_{{2,2}}a_{{3,2}}&-
a_{{2,3}}a_{{3,2}}+1-a_{{2,2}}a_{{3,3}}&-a_{{2,3}}a_{{3,3}}
\\ -{a_{{3,1}}}^{2}&-2\,a_{{3,2}}a_{{3,1}}&-2\,a_{{3
,3}}a_{{3,1}}&-{a_{{3,2}}}^{2}&-2\,a_{{3,3}}a_{{3,2}}&1-{a_{{3,3}}}^{2
}}
$$
In order to have a nonzero solution, the determinant of $B$ must be $0$.
This is a rather complicated polynomial of total degree 12, which factors over the rationals as
$$
- ( a_{{2,3}}a_{{3,1}}a_{{1,2}}-a_{{1,2}}a_{{2,1}}-a_{{2,1}}a_{{1
,2}}a_{{3,3}}+1+a_{{1,1}}+a_{{2,2}}+a_{{1,1}}a_{{2,2}}-a_{{1,3}}a_{{3,
1}}-a_{{2,2}}a_{{3,1}}a_{{1,3}}+a_{{2,1}}a_{{3,2}}a_{{1,3}}-a_{{2,3}}a
_{{3,2}}-a_{{1,1}}a_{{2,3}}a_{{3,2}}+a_{{3,3}}+a_{{1,1}}a_{{3,3}}+a_{{
2,2}}a_{{3,3}}+a_{{1,1}}a_{{2,2}}a_{{3,3}} ) ( a_{{2,3}}a_
{{3,1}}a_{{1,2}}+a_{{1,2}}a_{{2,1}}-a_{{2,1}}a_{{1,2}}a_{{3,3}}-1+a_{{
1,1}}+a_{{2,2}}-a_{{1,1}}a_{{2,2}}+a_{{1,3}}a_{{3,1}}-a_{{2,2}}a_{{3,1
}}a_{{1,3}}+a_{{2,1}}a_{{3,2}}a_{{1,3}}+a_{{2,3}}a_{{3,2}}-a_{{1,1}}a_
{{2,3}}a_{{3,2}}+a_{{3,3}}-a_{{1,1}}a_{{3,3}}-a_{{2,2}}a_{{3,3}}+a_{{1
,1}}a_{{2,2}}a_{{3,3}} ) ( -1-a_{{1,3}}a_{{3,1}}-2\,a_{{1,
2}}a_{{1,3}}{a_{{2,1}}}^{2}a_{{3,2}}a_{{3,3}}-2\,{a_{{1,2}}}^{2}a_{{2,
1}}a_{{2,3}}a_{{3,1}}a_{{3,3}}-2\,a_{{2,1}}{a_{{1,3}}}^{2}a_{{2,2}}a_{
{3,1}}a_{{3,2}}-2\,a_{{1,1}}a_{{1,2}}{a_{{2,3}}}^{2}a_{{3,1}}a_{{3,2}}
-2\,a_{{1,1}}a_{{1,3}}a_{{2,1}}a_{{2,3}}{a_{{3,2}}}^{2}-2\,a_{{1,2}}a_
{{1,3}}a_{{2,2}}a_{{2,3}}{a_{{3,1}}}^{2}-a_{{1,1}}a_{{1,3}}a_{{2,1}}a_
{{3,2}}+a_{{1,1}}a_{{1,3}}a_{{2,2}}a_{{3,1}}+a_{{2,3}}a_{{2,2}}a_{{1,1
}}a_{{3,2}}+a_{{2,1}}a_{{1,2}}a_{{1,1}}a_{{3,3}}+a_{{2,2}}a_{{2,1}}a_{
{1,2}}a_{{3,3}}-a_{{2,3}}a_{{3,1}}a_{{1,2}}a_{{3,3}}+a_{{2,2}}a_{{3,1}
}a_{{1,3}}a_{{3,3}}-a_{{2,1}}a_{{3,2}}a_{{1,3}}a_{{3,3}}+a_{{1,1}}a_{{
2,3}}a_{{3,2}}a_{{3,3}}-a_{{2,1}}a_{{2,2}}a_{{3,2}}a_{{1,3}}-a_{{2,2}}
a_{{2,3}}a_{{3,1}}a_{{1,2}}-a_{{1,1}}a_{{1,2}}a_{{2,3}}a_{{3,1}}+a_{{1
,3}}{a_{{2,2}}}^{2}a_{{3,1}}+{a_{{1,1}}}^{2}a_{{2,3}}a_{{3,2}}-a_{{2,2
}}{a_{{3,3}}}^{2}a_{{1,1}}+{a_{{1,1}}}^{2}{a_{{2,2}}}^{2}{a_{{3,3}}}^{
2}-{a_{{1,1}}}^{2}a_{{2,2}}a_{{3,3}}+{a_{{1,1}}}^{2}{a_{{2,3}}}^{2}{a_
{{3,2}}}^{2}-a_{{1,1}}{a_{{2,2}}}^{2}a_{{3,3}}+{a_{{1,2}}}^{2}{a_{{2,3
}}}^{2}{a_{{3,1}}}^{2}+{a_{{1,3}}}^{2}{a_{{2,2}}}^{2}{a_{{3,1}}}^{2}-2
\,a_{{1,1}}a_{{1,2}}a_{{2,1}}a_{{2,2}}{a_{{3,3}}}^{2}+{a_{{1,2}}}^{2}{
a_{{2,1}}}^{2}{a_{{3,3}}}^{2}+a_{{2,1}}{a_{{3,3}}}^{2}a_{{1,2}}+{a_{{1
,3}}}^{2}{a_{{3,2}}}^{2}{a_{{2,1}}}^{2}+2\,a_{{1,2}}a_{{2,3}}a_{{3,2}}
a_{{2,1}}a_{{1,1}}a_{{3,3}}-2\,{a_{{2,2}}}^{2}a_{{3,3}}a_{{1,3}}a_{{1,
1}}a_{{3,1}}+2\,a_{{2,1}}a_{{2,2}}a_{{3,3}}a_{{1,3}}a_{{1,2}}a_{{3,1}}
+2\,a_{{1,3}}a_{{3,2}}a_{{2,3}}a_{{2,2}}a_{{1,1}}a_{{3,1}}+2\,a_{{1,2}
}a_{{1,3}}a_{{2,1}}a_{{2,3}}a_{{3,1}}a_{{3,2}}+2\,a_{{1,1}}a_{{1,2}}a_
{{2,2}}a_{{2,3}}a_{{3,1}}a_{{3,3}}+2\,a_{{1,1}}a_{{1,3}}a_{{2,1}}a_{{2
,2}}a_{{3,2}}a_{{3,3}}-2\,{a_{{1,1}}}^{2}a_{{2,2}}a_{{2,3}}a_{{3,2}}a_
{{3,3}}-a_{{1,2}}a_{{2,1}}+a_{{1,1}}a_{{2,2}}+a_{{1,1}}a_{{3,3}}-a_{{2
,3}}a_{{3,2}}+a_{{2,2}}a_{{3,3}} )
$$