Theorems V in [this paper][1] of [L.E. Dickson][2] states that the following two sets are equal. $$E=\{a^2+b^2+2c^2 \ | \ a,b,c \in \mathbb{Z}\} \  \text{ and } \ F=\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \in \mathbb{N}\}.$$
Let $\mathbb{A}$ denote $\mathbb{N}$ or $\mathbb{Z}$. Consider the following set (where $u \cdot v$ denotes the usual [dot product][3]):   $$E(\mathbb{A}) = \{\|u \|^2 + |u \cdot v| \text{ such that } u,v \in \mathbb{A}^3 \text{ and } \|u \| = \|v \|\}.$$

Let $u= (a,b,c)$, $v = (b,-a,c) \in \mathbb{Z}^3$. Then $\|u \| = \|v \|$ and $\|u \|^2 + |u \cdot v| = a^2+b^2+2c^2.$
It follows that $F= E \subseteq E(\mathbb{Z})$.   

In fact, Dickson's theorem extends to $E(\mathbb{Z})$, since Philipp Lamp shown below that $E(\mathbb{Z}) = F$ also (as an answer to what was **Question 1** in a previous version).  

The computation below suggests the following question (checked for integers less than $5936$). 

> **Question 2**: Is it true that $E(\mathbb{N}) = F \setminus \{ 5, 23, 29, 65, 167 \} $?

*Application*: [this answer][4] proves that the form $\| A\|^2$ covers every natural number for $A \in M_3(\mathbb{Z})$.  
 A positive answer to Question 2 would prove this result for $A \in M_3(\mathbb{N})$.  

___

*Reformulation of Question 2* 

Take $u=v \in \mathbb{N}^3$, then $\|u \|^2 + |u \cdot v| = 2 \|u \|^2$, so by [Legendre's three-square theorem][5], $$2\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \in \mathbb{N}\} \subset E(\mathbb{N}).$$ 
So we are reduced to prove that 
$$2\mathbb{N}+1 \setminus \{ 5, 23, 29, 65, 167 \} \subset E(\mathbb{N}).$$

Now, as pointed out by Philipp Lampe, if $\|u \| = \|v \|$ then $\|u \|^2 + |u \cdot v| = \|u+v \|^2/2$.   
Then Question 2 can be reformulated as follows:  

> **Reformulated question**: Is it true that, for $u,v \in \mathbb{N}^3$ with $\|u \| = \|v \|$, the form $\|u+v \|^2/2$ covers every odd
> number, except those in $\{ 5, 23, 29, 65, 167 \}$?


___
**Computation**   


    sage: L=[]
    ....: for a1 in range(50):
    ....:     for a2 in range(a1+1):
    ....:         for a3 in range(a2+1):
    ....:             x=a1**2+a2**2+a3**2
    ....:             b=0
    ....:             while b<50 and b**2<x:
    ....:                 b+=1
    ....:             for b1 in range(b+1):
    ....:                 bb=0
    ....:                 while bb<50 and bb**2<x-b1**2:
    ....:                     bb+=1
    ....:                 for b2 in range(bb+1):
    ....:                     bbb=0
    ....:                     while bbb<50 and bbb**2<x-b1**2-b2**2:
    ....:                         bbb+=1
    ....:                     for b3 in range(bbb+1):
    ....:                         if a1**2+a2**2+a3**2==b1**2+b2**2+b3**2:
    ....:                             n=((a1+b1)**2+(a2+b2)**2+(a3+b3)**2)/2
    ....:                             L.append(n)
    ....: l=list(set(L)); l.sort()
    ....: s=set(range(5936))-set(l)
    ....: S=[]
    ....: for i in s:
    ....:     f=list(factor(i))
    ....:     a=f[0][0]
    ....:     b=f[0][1]
    ....:     if a<>2:
    ....:         S.append(i)
    ....:     elif Integer(b).mod(2)==0:
    ....:         S.append(i)
    ....:     elif Integer(i/(2**b)).mod(8)<>7:
    ....:         S.append(i)
    ....: S.sort()
    ....: S
    [5, 23, 29, 65, 167]


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=1561323
  [2]: https://en.wikipedia.org/wiki/Leonard_Eugene_Dickson
  [3]: https://en.wikipedia.org/wiki/Dot_product
  [4]: https://math.stackexchange.com/a/2967989/84284
  [5]: https://en.wikipedia.org/wiki/Legendre%27s_three-square_theorem