Interval Preserving in Minkowski Space The squared line element in any spacetime is given as $$ds^{2}=g_{ab}dx^{a}dx^{b}.$$ The use of tensors helps us to infer that the line element in some other frame would be $$ds'^{2}=g'_{ab}dx'^{a}dx'^{b}$$ where simply $dx'^{a}=\frac{\partial x'^{a}}{\partial x^{b}} dx'^{b}$. 
My question is, in special relativity, there is further a condition on the line element that it should $$c^{2}(s-t)^{2}-(x_{1}-y_{1})^{2}-(x_{2}-y_{2})^{2}-(x_{3}-y_{3})^{2}=c^{2}(s'-t')^{2}-(x'_{1}-y'_{1})^{2}-(x'_{2}-y'_{2})^{2}-(x'_{3}-y'_{3})^{2}$$ which gives us the Lorentz transformations. How can we prove this condition using the postulates of special relativity?
Also where and how do we employ the condition that the frames we are transforming to are inertial?
 A: In fact THIS IS a postulate, the second postulate of Einsteins's special relativity, velocity of light must be invariant in any inertial frame of reference, so mathematically, for a light wave we have : $cdt^2-dx^2-dy^2-dz^2=0$ and $cdt'^2-dx'^2-dy'^2-dz'^2=0$ in another inertial frame, and so $ds^2=ds'^2$ ; this is the origin of your 'further condition'. General relativity agree with this just in local frames.
A: Here why $\eta_{ab} = \eta_{mn} \Lambda^m_a \Lambda^n_b$ in SR (special relativity).
$ds^2 = \eta_{ab} dx^a dx^b$ (1)
$ds^2 = \eta_{mn} dx^m dx^n$ (2)
but:
$dx^m = \Lambda^m_a dx^a$ Lorentz transformation
so:
$\eta_{mn} dx^m dx^n = \eta_{mn} \Lambda^m_a \Lambda^n_b dx^a dx^b$ (3)
by equating (1) and (2), taking count of (3):
$\eta_{ab}=\eta_{mn}\Lambda_a^m\Lambda_b^n$
Note:
In SR you have linear transformations as Lorentz transformation is linear in the coordinates as a consequence of the principles of special relativity.
A: The reason you've been unable to find a derivation of the Lorentz transformation (relating, say, Bob's frame to Alice's) from the usual  two postulates of special relativity is that the Lorentz transformation does not follow from those postulates.  You're going to need some additional assumption.
You can, for example, add some assumption about homogenenity/isotropy (though it's not enough, of course, to just mumble the words "homogeneity" and "isotropy"; you need some careful statement of exactly what you're assuming), or you can start by assuming that Bob's measurement of Alice's velocity is constant over time.   Or you can just assume  that the transformation must be linear.  
There might then be room for someone to argue that your new assumption is really just a consequence of the old assumptions (in particular that the laws of nature "look the same" to both Bob and Alice).  Since the old assumption is pretty vague to begin with (never specifying exactly what counts as a law of nature), it's possible to defend this argument, but it's at least a stretch.
