= Solution
Use coordinates $X=(ct,x,y,z)^T$ and <Minkowski metric> $\eta=\operatorname{diag}(1,-1,-1,-1)$. A real $4\times4$ <matrix> $L$ is a <Lorentz transformation> precisely when it preserves the <Minkowski interval> for every $X$, equivalently
$$
\boxed{L^T\eta L=\eta.}
$$
It is proper when $\boxed{\det L=+1}$, and orthochronous when $\boxed{L^0{}_0>0}$, equivalently $L^0{}_0\ge1$. The latter selects the <orthochronous Lorentz group>, which preserves future-directed timelike <four-vectors>. Indeed the time-time entry of the preservation identity gives $(L^0{}_0)^2-\sum_{i=1}^3(L^i{}_0)^2=1$, so $|L^0{}_0|\ge1$. Imposing both conditions gives a <proper orthochronous Lorentz transformation>.
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