I saw that various quantities that were of the form: $\textbf{a}\eta\textbf{b}$, where $\textbf{a}$ and $\textbf{b}$ are some four vectors, are Lorentz invariant. For example, it seems that $p_t^2-p_x^2-p_y^2-p_z^2$ is lorentz invariant and so is $ctp_t-xp_x-yp_y-zp_z$. So can we extend this to the general case for any 2 four vectors $\textbf{a,b}$?
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Yes, the two examples you've given are the contraction of the four-momentum with itself:
$$\eta_{\mu\nu}p^\mu p^\nu=p_\nu p^\nu \tag{1},$$ and the contraction of the four-position with the four-momentum:
$$\eta_{\mu\nu}x^\mu p^\nu=x_\nu p^\nu \tag{2}.$$
These are specific examples of how we build invariant quantities in special relativity. The four-position and four-momentum have different components in different inertial frames (which corresponds to different inertial coordinate systems on Minkowski space), but the scalar quantities obtained by contracting two Lorentz four-vectors together with the metric does not change.
In your first example the invariant quantity obtained from the contraction of the four-momentum with itself is the rest mass, which is a frame-invariant quantity. The second example does not have a standard physical interpretation to my knowledge, but it is still a coordinate invariant quantity by its definition.
Just to give further examples, the invariant quantity obtained from contracting the four-position with itself is the invariant spacetime interval, and that of the four-velocity is the speed of light. Both are independent of the inertial frame in which they are measured.
