Are these sums of squares identities new? The identities are
\begin{align}
 u^2 + \left( (2u-1)(2u^2-u+1) \right)^2 + \left( u (2u-1)(2u^2-u+1) \right)^2 &= \left( 4u^4-4u^3+5u^2-3u+1 \right)^2 \\
u^2 + \left( (2u+1)(2u^2+u+1) \right)^2 + \left( u (2u+1)(2u^2+u+1) \right)^2 &= \left( 4u^4+4u^3+5u^2+3u+1 \right)^2
\end{align}
I have not been able to find them online or in any of my books.

The complete solution to $x^2 + y^2 + z^2 = w^2$ is known: $$x = (a^2-b^2-c^2)t,\; y = 2 a b t,\; z = 2 a c t,\; w = (a^2 + b^2 + c^2) t$$
Your first  is the case
$$ \eqalign{a &= (2 u^2 - u + 1) \cr
            b &= 2u-1\cr
            c &= u (2u-1)\cr
            t &= \frac{1}{2}\cr }$$
Similarly for the second, with some sign changes.
