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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 $j \geq 2,$ j,$($j$can be odd or even) then $$\cos \theta = \frac{m}{2^j} frac{m}{4j} = \frac{m}{\sqrt{4^j}} frac{m}{\sqrt{16 j^2}}$$ 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. 4 added 264 characters in body 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$and$j \geq 2,$then $$\cos \theta = \frac{m}{2^j} = \frac{m}{\sqrt{4^j}}$$ 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. 3 added 148 characters in body 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.\$

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.

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