= Conformal Killing algebra of Minkowski spacetime
{title2=$\dim\mathfrak{conf}(\mathbb R^{1,3})=15$}
In <Cartesian coordinates> on four-dimensional <Minkowski spacetime>, all <Conformal Killing vector fields> have the form
$$
X^\mu=a^\mu+\omega^\mu{}_{\nu}x^\nu+cx^\mu+2(b\cdot x)x^\mu-x^2b^\mu,
\qquad \omega_{\mu\nu}=-\omega_{\nu\mu},\qquad x^2=\eta_{\mu\nu}x^\mu x^\nu.
$$
Their <conformal Killing factor> is $c+2b\cdot x$. There are four <translation> generators, six <Lorentz transformation> generators, one <dilation> generator, and four <special conformal transformation> generators. These form a fifteen-dimensional <Lie algebra> under the <Lie bracket of vector fields>.
For completeness, differentiate the <Conformal Killing equation> in three index orders to obtain
$$
\partial_\mu\partial_\nu X_\rho
=\eta_{\nu\rho}\partial_\mu\lambda+\eta_{\mu\rho}\partial_\nu\lambda-\eta_{\mu\nu}\partial_\rho\lambda.
$$
Its <divergence> gives $2\partial_\mu\partial_\nu\lambda=-\eta_{\mu\nu}\Box\lambda$; its <tensor contraction> gives $\Box\lambda=0$. Hence $\lambda$ is an <affine function>. Subtracting the displayed quadratic and <dilation> terms leaves a <Killing vector field> whose second <partial derivatives> vanish, proving completeness of the list. The <vector fields> are globally smooth, but some <special conformal transformations> have incomplete flows on affine <Minkowski spacetime>; the <dilation> flow $x\mapsto e^s x$ is global.
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