= Solution
{c}
Write the <affine function> obtained in part (iii) as $\lambda=c+2b_\mu x^\mu$, with constant $c$ and $b_\mu$. A <Conformal Killing vector field> with precisely this <conformal Killing factor> is
$$
X_{\mathrm{part}}^\mu=cx^\mu+2(b\cdot x)x^\mu-x^2b^\mu,
\qquad x^2=\eta_{\alpha\beta}x^\alpha x^\beta.
$$
Direct <partial differentiation> verifies its <Conformal Killing equation>. Subtracting it from any other solution leaves a <Killing vector field> $Y$. The formula in part (iii), now with $\lambda=0$, says $\partial_\mu\partial_\nu Y_\rho=0$. Therefore $Y$ is affine, and the <Killing equation> makes its linear coefficient antisymmetric after lowering an index. Every solution is consequently
$$
\boxed{X^\mu=a^\mu+\omega^\mu{}_{\nu}x^\nu+cx^\mu+2(b\cdot x)x^\mu-x^2b^\mu,
\qquad \omega_{\mu\nu}=-\omega_{\nu\mu}.}
$$
The four constant $a^\mu$ generate <translations>, the six independent $\omega_{\mu\nu}$ generate <Lorentz transformations>, $c$ generates a <dilation>, and the four $b^\mu$ generate <special conformal transformations>. Their constant, linear and quadratic terms are independent; within the linear terms, the antisymmetric coefficient cannot equal a nonzero multiple of the identity. Hence the <conformal Killing algebra of Minkowski spacetime> has
$$
\boxed{\dim\mathfrak{conf}(\mathbb R^{1,3})=4+6+1+4=15.}
$$
For $\lambda=1$, the canonical choice is the <dilation> generator $D=x^\mu\partial_\mu$. Its <flow of a vector field> solves $dx^\mu/ds=x^\mu$, giving
$$
\boxed{\Phi_s(x)=e^s x,\qquad \Phi_s^*\eta=e^{2s}\eta,\qquad \Phi_s\circ\Phi_t=\Phi_{s+t}.}
$$
One may add any <Killing vector field> without changing $\lambda=1$, so this choice is not unique. The displayed <dilation> gives a global <one-parameter group> of <diffeomorphisms>. The count of fifteen concerns independent <Conformal Killing vector fields>; some <special conformal transformations> have only local flows on all of affine <Minkowski spacetime>.
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