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• The first identity in this passage was verifed algebrically by me, and I also suspect it might be intimately connected with quaternions.
• The second identity follows directly from the first by substituting $\lambda = 1-i$ and $\mu = 1+i$. Since $N(1-i)+N(1+i)=4$, this identity enables one to generate new representations of an integer $4s$ as sum of four squares by simply changing the signs and order of the different numbers in the representation of $s$ as sum of four squares. For example, if $s = 13 = 2^2+2^2+2^2+1^2$ than this identity implies $52=4s = 5^2+3^2+3^2+3^2$.

• The first identity in this passage is intimately connected with quaternions - actually it is a kind of pre Cayley-Dickson construction, since Gauss essentially says here that the quaternions multiplication rules arise from the rules of complex arithmetic and can be constructed by them. To see this more clearly, lets make the following steps:

$$Nl+Nm=N(l+mj)=N(l-mj)$$ $$N\lambda+N\mu = N(\lambda+\mu j)=N(\lambda+j\mu)$$

Since the quaternions algebra is a composition algebra, the norm is multiplicative, so: $$N(l-mj)\cdot N(\lambda+j\mu) = N((l-mj)\cdot(\lambda+j\mu)) = N((l\lambda -(mj)(j\mu))+l(j\mu)-(mj)\lambda) = N((l\lambda+m\mu))+l(\bar{\mu}j)-m(\bar{\lambda}j)) = N((l\lambda+m\mu))+(l\bar{\mu}-m\bar{\lambda})j) = N(l\lambda+m\mu)+N(l\bar{\mu}-m\bar{\lambda})$$

Here the associativity of quaternions is used, as well as the fact that (for example) $j\mu = \bar{\mu}j$. For more on Gauss's anticipation of the quaternions algebra, look at the (partially answered) post Motivating unpublished statements of Gauss about congruences and quaternions.

• The second identity follows directly from the first by substituting $\lambda = 1-i$ and $\mu = 1+i$. Since $N(1-i)+N(1+i)=4$, this identity enables one to generate new representations of an integer $4s$ as sum of four squares by simply changing the signs and order of the different numbers in the representation of $s$ as sum of four squares. For example, if $s = 13 = 2^2+2^2+2^2+1^2$ than this identity implies $52=4s = 5^2+3^2+3^2+3^2$.

• The result of Gauss on the correspondence between the representation of a multiple of a prime number $p$ as sum of four squares and the solution to the congruence $x^2+y^2+z^2\equiv 0 \pmod{p}$ is straitforward to prove: $$x^2+y^2+z^2 = (a^2+b^2)^2+(ac+bd)^2+(ad-bc)^2=(a^2+b^2)^2+(a^2+b^2)(c^2+d^2)= (a^2+b^2)(a^2+b^2+c^2+d^2)\equiv 0 \pmod{p}$$. Therefore, what remains to be settled is the other results mentioned in remark 2.
• The result on the correspondence between the solutions $(x,y)$ of the congruence $1+x^2+y^2\equiv 0 \pmod{p}$ and the solution of a certain imaginary congruence of degree $p+1$ was also checked by me by taking specific examples: for example, if $p=7$, than $x = 3, y = 2$ is a solution, and $1+(3+2i)^8 = -238-28560i = 7\cdot(-34-4080i)$ is a Gaussian integer multiple of $7$.

• The result of Gauss on the correspondence between the representation of a multiple of a prime number $p$ as sum of four squares and the solution to the congruence $x^2+y^2+z^2\equiv 0 \pmod{p}$ is straitforward to prove: $$x^2+y^2+z^2 = (a^2+b^2)^2+(ac+bd)^2+(ad-bc)^2=(a^2+b^2)^2+(a^2+b^2)(c^2+d^2)= (a^2+b^2)(a^2+b^2+c^2+d^2)\equiv 0 \pmod{p}$$. Therefore, what remains to be settled is the other results mentioned in remark 2.

• The result on the correspondence between the solutions $(x,y)$ of the congruence $1+x^2+y^2\equiv 0 \pmod{p}$ and the solution of a certain imaginary congruence of degree $p+1$ was checked by me by taking specific examples: for example, if $p=7$, than $x = 3, y = 2$ is a solution, and $1+(3+2i)^8 = -238-28560i = 7\cdot(-34-4080i)$ is a Gaussian integer multiple of $7$.

• The papers "On the Computation of Representations of Primes as Sums of Four Squares" and "The circle equation over finite fields" mention results equivalent to Gauss's results in this remark. In particular, Gauss's method of generating new solutions to the congruence $1+x^2+y^2\equiv 0 \pmod{p}$ by $x+iy = (x_0+iy_0)(\frac{u+i}{u-i})$ is refered to as "the method of diophantus" in section 2.2 of the first paper i mentioned. Since both papers appear not to be very advanced, i believe that explaning the results in remark 2 is an easy task in the standards of mathoverflow.

• The result of Gauss on the correspondence between the solutions $(x,y)$ of the congruence $1+x^2+y^2\equiv 0 \pmod{p}$ and the solution of a certain imaginary congruence of degree $p+1$ was checked by me by taking specific examples: for example, if $p=7$, than $x = 3, y = 2$ is a solution, and $1+(3+2i)^8 = -238-28560i = 7\cdot(-34-4080i)$ is a Gaussian integer multiple of $7$.

• The two propositions which Gauss mentions in remark 1 are unclear to me, and I mean that the propositions themself are unclear, not their derivation. It is simply not formulated clearly. Therefore, I'd like to get an explanation of the two propositions which Gauss mentions, as well as an explanation of its proof.
• I'd like to understand how the theta function identity in remark 1 follows from the two propositions.
• What is the expalanation of the results in remark 2? they are very complicated and i don't have a clue of understanding its proof.

• The result of Gauss on the correspondence between the representation of a multiple of a prime number $p$ as sum of four squares and the solution to the congruence $x^2+y^2+z^2\equiv 0 \pmod{p}$ is straitforward to prove: $$x^2+y^2+z^2 = (a^2+b^2)^2+(ac+bd)^2+(ad-bc)^2=(a^2+b^2)^2+(a^2+b^2)(c^2+d^2)= (a^2+b^2)(a^2+b^2+c^2+d^2)\equiv 0 \pmod{p}$$. Therefore, what remains to be settled is the other results mentioned in remark 2.
• The result on the correspondence between the solutions $(x,y)$ of the congruence $1+x^2+y^2\equiv 0 \pmod{p}$ and the solution of a certain imaginary congruence of degree $p+1$ was also checked by me by taking specific examples: for example, if $p=7$, than $x = 3, y = 2$ is a solution, and $1+(3+2i)^8 = -238-28560i = 7\cdot(-34-4080i)$ is a Gaussian integer multiple of $7$.

• The two propositions which Gauss mentions in remark 1 are unclear to me, and I mean that the propositions themself are unclear, not their derivation. It is simply not formulated clearly. Therefore, I'd like to get an explanation of the two propositions which Gauss mentions, as well as an explanation of its proof.
• I'd like to understand how the theta function identity in remark 1 follows from the two propositions.
• What is the expalanation of the results in remark 2? they are complicated and i don't have a clue of understanding its proof.