Axis decomposition of a symmetric matrix ($T=AI+Bv{v}^T+C{v}^T+v{C}^T+D$)

Fix $v\ne 0$, put $s=v^{T}v$, $t=v^{T}Tv$, and $P=I-v{v}^T/s$. Any real symmetric matrix on three-dimensional Euclidean space has a unique decomposition with $C^{T}v=0$, $Dv=0$ and $\mathrm{tr}D=0$:The transverse restriction is a symmetric two-dimensional matrix, whose traceless matrix part has dimension two. Together the two scalars, two components of $C$ and two components of $D$ give the six parameters of $T$.