The general properties of spacetime intervals arise from their general definition, which refers to geometric relations (distances, durations) between members of inertial systems (of course without referring to any incidental coordinate assignments):
timelike intervals: If the same one member $A$ of an inertial system had taken part (successively) in two distinct events $\varepsilon_{A~P}$ (i.e. the event of $A$ and a suitable participant $P$ having been coincident, having met each other in passing) and $\varepsilon_{A~Q}$ (i.e. the event of $A$ and a suitable participant $Q$ having been coincident, having met each other in passing)
then the (value of the) spacetime interval between these two events, $s^2[~\varepsilon_{A~P}, \varepsilon_{A~Q}~]$, is defined as proportional to the square of participant $A$'s duration between having indicated the coincidence with $P$ and having indicated the coincidence with $Q$: $$ s^2[~\varepsilon_{A~P}, \varepsilon_{A~Q}~] := -c^2~\tau A[~{}_P, {}_Q~],$$
where $-c^2$ is a conventional non-zero proportionality factor.
(If $\varepsilon_{A~P}$ and $\varepsilon_{A~Q}$ are actually one and the same event then $A$'s corresponding duration is of course Null, and so is therefore the spacetime interval between one event and the identical same event.)
Considering the geometric relations between participant $A$ and members of another inertial frame (to which $A$ does not belong), say $F$ and $G$ (who were and remained at rest wrt. each other), such that
- $F$ also took part in event $\varepsilon_{A~P} \equiv \varepsilon_{A~F~P}$, and
- $G$ also took part in event $\varepsilon_{A~Q} \equiv \varepsilon_{A~G~Q}$,
then (as has been proven separately, e.g. here)
$$\tau A[~{}_P, {}_Q~] = \sqrt{ (\tau G[~{}_{AQ}, {}_{\circledS F \circ AP}~])^2 - \left(\frac{1}{c}~d[~F, G~]\right)^2 } = \tau G[~{}_{AQ}, {}_{\circledS F \circ AP}~]~\sqrt{1 - (\beta_{FG}[~A~])^2}, \tag{1}$$
where $\tau G[~{}_{AQ}, {}_{\circledS F \circ AP}~]$ is the duration of participant $G$ from having indicated the coincidence with $A$ and $Q$ until $F$'s indication simultaneous to $F$'s indication of coincidence with $A$ and $P$,
$d[~F, G~]$ is the distance between $F$ and $G$, and
$\beta_{FG}[~A~]~c$ is the speed of $A$ wrt. the inertial system of which $F$ and $G$ are members.
On the other hand:
spacelike intervals: Considering two members of the same inertial frame (i.e. two participants who were and remained at rest wrt. each other), $A$ and $B$, where $A$ took part in one event, $\varepsilon_{A~J}$, and $B$ took part in another event $\varepsilon_{A~K}$, and if the corresponding indications of $A$ and of $B$, resp., at these events, namely indications $A_J$ and $B_K$ were simultaneous to each other, then
then the (value of the) spacetime interval between these two events, $s^2[~\varepsilon_{A~J}, \varepsilon_{B~K}~]$, is defined as proportional to the square of the (chronometric) distance between $A$ and $B$:
$$ s^2[~\varepsilon_{A~J}, \varepsilon_{B~K}~] := \alpha~d[~A, B~],$$
where the common non-zero constant of proportionality is still to be determined, in relation to the above convention.
(Again, and consistently with the conclusion reached above already, if $\varepsilon_{A~J}$ and $\varepsilon_{B~K}$ are actually one and the same event, such that $A$ and $B$ are in fact names referring to one and the same participant then $A$'s corresponding distance is of course Null, and so is therefore the spacetime interval between one event and the identical same event.)
Considering the geometric relations between participants $A$ and $B$ and members of another inertial frame (to which $A$ and $B$ don't not belong), say $F$ and $G$ (who were and remained at rest wrt. each other), such that
- $F$ also took part in event $\varepsilon_{A~J} \equiv \varepsilon_{A~F~J}$, and
- $G$ also took part in event $\varepsilon_{B~K} \equiv \varepsilon_{B~G~K}$, where (for definiteness in the following) $G$'s indication of coincidence with $B$ and $K$ was before $G$'s indication simultaneous to $F$'s indication of coincidence with $A$ and $J$,
then (as has been proven separately)
$$d[~A, B~] = \sqrt{ (d[~F, G~])^2 - c^2~(\tau G[~{}_{BK}, {}_{\circledS P \circ AJ}~])^2 } = d[~F, G~]~\sqrt{1 - (\beta_{FG}[~B~])^2}. \tag{2}$$
The formal similarity between eqs. (1) and (2) suggests to set $\alpha := 1$ and to arrive at the formal, generalized expression for interval values in terms of geometric relations between members of a general inertial frame:
$$s^2[~\text{event in which }F\text{ took part}, \text{event in which }G\text{ took part}~] := -c^2~(\tau G[~{\small{\text{indication of own participation}, \text{indication simultaneous to }F\text{'s indication of participation}}}~])^2 + (d[~F, G~])^2, \tag{3}$$
or reduced to a mere mnemonic:
$$s^2 := -c^2~\text{duration}^2 + \text{distance}^2.$$
Finally,
lightlike intervals: if neither of the two cases discussed above applies, but instead participants in one event observed the signalfront of (indications of participants having taken part in) the other event, then the corresponding interval value between these two events is defined as Null:
$$s^2[~\text{pair of lightlike related events}~] = 0.$$
This is also consistent with the formal expression~(3), with the chronometric distance between any two members of one inertial system, such as participants $F$ and $G$, being proportional to the ping duration of either one wrt. the other.
