Relativistic sum with magnitude c 
Pick any two vectors (in 3 dimensions) having magnitude equal to c and check whether the relativistic sum of them also has magnitude c. Is u v equal to v u?

Let $ ||u||_2 = ||v||_2 = c$. Then:
\begin{align}
||u \oplus v||_2 &= \frac{1}{c^2 + u\cdot v}\left\vert\left\vert
 c^2(u+v) 
+ 
\frac{u\times(u\times v)}{1 + \sqrt{1-(u\cdot u)/c^2}} \right\vert\right\vert_2 \\
&=
\frac{1}{c^2 + u\cdot v}
|| 
\underbrace{c^2(u+v) }_a
+
\underbrace{ u\times(u\times v)  }_b
||_2 \\
&= \frac{1}{c^2 + u\cdot v} \sqrt{ a\cdot a + b\cdot b + 2(a\cdot b) } \\
\end{align}
Expanding these dot products:
\begin{align}
a\cdot a 
&=
||c^2(u+v)||_2^2 \\
&=
c^4[(u\cdot u) + (v\cdot v) + 2(u\cdot v)] \\
&= 2c^4[c^2 + (u\cdot v)]\\
b\cdot b 
&=
||u\times(u \times v)||_2 \\
&=
||u||_2^2 ||u\times v||_2^2 - \underbrace{[u\cdot (u\times v)]^2}_0 \\
&=
c^2[||u||_2^2 ||v||_2^2 - (u\cdot v)^2] \\
&= c^2[c^4 - (u\cdot v)^2] \\
a\cdot b
&=
c^2[(u+v)\cdot (u\times(u\times v))] \\
&= c^2[ \underbrace{u \cdot (u\times(u\times v))}_0 + v \cdot (u\times(u\times v)) ] \\
&= c^2[(u\times v)\cdot (v\times u)] \\
&= -||u\times v||_2^2 \\
&= -[c^4 - (u\cdot v)^2]
\end{align}
Plugging back in:
\begin{align}
||u \oplus v||_2
&=
\frac{1}{c^2 + u\cdot v} \sqrt{
2c^4[c^2 + (u\cdot v)]
+
c^2[c^4 - (u\cdot v)^2] 
-2[c^4 - (u\cdot v)^2]c^2
} \\
&=
\frac{c}{c^2 + u\cdot v}
\sqrt{ c^4 + 2(u\cdot v) c^2 + (u\cdot v)^2 } \\
&=
\frac{c}{c^2 + u\cdot v}
[c^2 + (u\cdot v)] \\
&= c
\end{align}
as expected.
As for the second part, when the norms are $c$, $u\oplus v = v\oplus u$, since $u$ and $v$ were arbitrary. 
