Louis J. Mordell showed in 1930 that the matrix equation
[

\left( \begin{array}{rrrr}
p & q & r & s\\\
t & u & v & w
\end{array}
\right)
\; \cdot \;
\left( \begin{array}{rr}
p & t \\\
q & u \\\
r & v \\\
s & w
\end{array}
\right) = \; \;\; \;
\left( \begin{array}{cc}
A & B \\\
B & C
\end{array}
\right)

, ]
all quantities integers, while the right hand side is positive definite, is possible if and only if the determinant
$A C - B^2$ is the sum of three squares, that is not of shape $4^k ( 8 n + 7).$

We are interested in the possible lattice angles in $\mathbf R^4,$ which we write with positive integers $F,G,$ (such that $F^2 < G$) as
$$ \cos \theta = \frac{\pm F}{\sqrt{G}}. $$ The main demand we make is that if $F$ is even, we require
that $G$ is not divisible by 4. We ask whether it is possible to realize this angle on the lattice.

When $F$ is odd, this is possible if and only if $G \neq 0 \pmod 8.$

When $F \equiv 2 \pmod 4,$ (so $G \neq 0 \pmod 4$), this is possible if and only if $G \neq 3 \pmod 8.$

When $F \equiv 0 \pmod 4,$ (so $G \neq 0 \pmod 4$), this is possible if and only if $G \neq 7 \pmod 8.$

Whenever possible, the solution is given by the 2 by 2 matrix above, with $B = F$ and $ A C = G.$
Then, as we have $G - F^2 \equiv 1,2,3,5,6 \pmod 8,$ once we select our favorite $A$ and $C,$ there are in fact integral vectors $P = (p,q,r,s)$ and $T = (t,u,v,w)$ such that
$$P \cdot P = A, \; \; \; P \cdot T = B, \; \; \; T \cdot T = C.$$

Note that we can actually take $A = 1, \; B = F, \; C = G, \; \; P = (1,0,0,0), \; \; T = (F,u,v,w)$ by demanding $u^2 + v^2 + w^2 = G - F^2.$

Also note that some rational values are ruled out. While
$ \cos \theta = 1/2$ is easy, if we have odd $m$ and any $j,$ ($j$ can be odd or even) then
$$ \cos \theta = \frac{m}{4j} = \frac{m}{\sqrt{16 j^2}} $$
cannot be arranged with $\theta$ a lattice angle in $\mathbf R^4.$

L. J. Mordell, "A new Waring's problem with squares of linear forms," Quarterly Journal of Mathematics (1930), pages 276-288. Seems to be volume 1.