The conformal group is the set of transformation that preserve angles. With this idea, then a conformal transformation is such that $x\rightarrow x^\prime$ and $$ g^\prime_{\mu\nu}(x^\prime) = \Omega(x)g_{\mu\nu}(x),\quad \Omega: \mathbb{R}^d\mapsto\mathbb{R} $$ with $d$ the spacetime dimension.
It can be proven that the whole set of conformal transformations is the sum of dilations, translations, the so-called special conformal transformations, and Lorentz transformations. These last ones comprise rotations and boosts.
Let's say our metric $g_{\mu\nu}$ is Minkowski's. We know that a boost produces the contraction of lengths in the direction of the boost. Let's consider a couple of vectors with same origin in the $yz$-plane and a boost in $y$-axis. Then, we would see a reduction on the $y$-components of these vectors while the $z$-components are preserved. This clearly modifies the angle between the vectors.
My question is: how is then the whole Lorentz group (not just the subgroup of rotations) part of the conformal group?