How to prove Lorentz invariance for rotations? I'm a bit stuck in the part of special relativity, as all the books I've read assume previous knowledge of the topic.
I would like to know how can I show that a quantity is invariant under Lorentz rotations and boosts. From my understanding with Lorentz boosts, I just have to prove that the quantity remains constant after applying the transformation.
For example, for the invariant:
$(ct)^2-x^2-y^2-z^2=(ct')^2-x'^2-y'^2-z'^2$
If I apply, for example, a Lorentz boost of the form:
$t'=\gamma(t-vx/c^2) \ \ , \ \ x'=x(\gamma-vt) \ \ , \ \ y'=y \ \ , \ \ z'=z$
It's just a matter of substituting and developing terms, so it's straightforward. 
However, for Lorentz rotations I'm not entirely sure how to proceed as I don't know how to build the rotation that is still Lorentz invariant (all the articles only show the case for Lorentz boosts).
 A: Rotations only change the spatial coordinates $(x^i)$; the time coordinate $(x^0)$ stays unchanged.
Now suppose you're rotating around the $z$ axis. Then the rotation matrix $(R)$ for this is:
\begin{equation}
R = 
\begin{bmatrix}
  \cos{\theta} & -\sin{\theta} & 0\\
  \sin{\theta} & \cos{\theta} & 0\\
  0 & 0 & 1
\end{bmatrix}
\end{equation}
This induces a rotation of coordinates $(x \rightarrow x')$ in component form as:
\begin{equation}
x'^i = R^i_{\ j}x^j
\end{equation}
Note that the $3 \times 3$ matrix above is the spatial part of the $4 \times 4$ Lorentz transformation matrix $\Lambda$.
Now to show Lorentz invariance under this special case of rotation around $z$ axis, we just need to show that $(x')^2 + (y')^2 = (x)^2 + (y)^2$, which is trivial.
A: Why a Rotation matrix $R$ don't "destroy" the Lorentz invariant
Prove what Mr. Avantgarde wrote
\begin{align*}
 & \text{General Lorentz Transformation matrix}\\\\ 
 \Lambda&=\begin{bmatrix}
            \gamma & \gamma \,\vec{v}^T \\
            \gamma \,\vec{v}^T & I+\frac{\gamma-1}{\vec{v}^2}\,\vec{v}\,\vec{v}^T \\
          \end{bmatrix}\\\\
&\text{with:}\\
\gamma&=\frac{1}{\sqrt{1-\frac{\vec{v}^2}{c^2}}}\\\\
\vec{v}&=\begin{bmatrix}
           v_x \\
           v_y \\
           v_z \\
         \end{bmatrix}\,,\quad\text{velocity boost vector}\\
I&=\begin{bmatrix}
     1 & 0 & 0 \\
     0 & 1 & 0 \\
     0 & 1 & 1 \\
   \end{bmatrix}\,,\quad\text{$3\times3$ identity  matrix}\\\\
&\text{Line}\\
   s^2&=x_\mu\,x^\mu=\eta_{\mu\nu}x^\nu\,x^\mu\\
   s^2&=\vec{x}^T\,\eta\,\vec{x}=(x^0)^2-(x^1)^2-(x^2)-(x^3)\,,\quad \text{with}\\
   \vec{x}&=\begin{bmatrix}
              x^0 \\
              \vec{x}^i \\
            \end{bmatrix}\,,\quad \text{and}\\
\eta&=\begin{bmatrix}
         1 & 0 & 0 & 0 \\
         0 & -1 & 0 & 0 \\
         0 & 0 & -1 & 0 \\
         0 & 0 & 0 & -1 \\
       \end{bmatrix}
   \end{align*}
I) Lorentz transformation: Pure boost
\begin{align*}
&\text{line}\\
s'^2&=x'_\mu\,x'^\mu\,,\quad\text{with}\quad x'^\mu=\Lambda\,\vec{x}\quad\text{and}\quad x'_\mu=\eta\,\vec{x}\,\quad\Rightarrow   \\
s'^2&=\vec{x}^T\,\underbrace{\Lambda^T\,\eta\,\Lambda}_{=\eta}\vec{x}=  \vec{x}^T\,\eta\,\vec{x}=s^2\,,\quad\text{Lorentz invariant}
\end{align*}
II) Lorentz transformation: Boost plus rotation matrix $R$ 
\begin{align*}
 \vec{x}&\mapsto 
 \begin{bmatrix}
              x^0 \\
              R\,\vec{x}^i \\
            \end{bmatrix}\\
\quad \Rightarrow\\
&\text{line}\\
s'^2&=\vec{x}^T\,\underbrace{\Lambda^T\,\eta\,\Lambda}_{=\eta},\vec{x}=  \vec{x}^T\,\eta\,\vec{x}=
\begin{bmatrix}
  x^0 &  \left(R\,\vec{x}^i\right)^T \\
\end{bmatrix}\begin{bmatrix}
         1 & 0 & 0 & 0 \\
         0 & -1 & 0 & 0 \\
         0 & 0 & -1 & 0 \\
         0 & 0 & 0 & -1 \\
       \end{bmatrix}
\begin{bmatrix}
  x^0 \\
  R\,\vec{x}^i \\
\end{bmatrix}\\
&=\begin{bmatrix}
  x^0 &  \vec{x^i}^T R^T \\
\end{bmatrix}
\begin{bmatrix}
  x^0 \\
  -R\,\vec{x}^i \\
\end{bmatrix}=(x^0)^2-(x^1)^2-(x^2)-(x^3)\,, \quad\text{again Lorentz invariant}\\
&\text{with}\quad R^T\,R=I
\end{align*}    
