Is there a generalization of Brahmagupta–Fibonacci identity for cubic? I was wondering if there was a cubic version of Brahmagupta–Fibonacci identity .
I looked everywhere but I didn't find anything ,except the splendid website of Tito Piezas (see here)
Furthermore I know the Gauss identity wich is :
$$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$
So my question is :
Is there a generalization of Brahmagupta–Fibonacci identity or Gauss identity ?
Thanks a lot .

There is, provided your are prepared to have 3 letters in the identity. It requires consideration of multiplication in a cubic field. See http://www.mathjournals.org/jrms/2017-032-004/2017-032-004-001.html 
Let $$N_j = \left ( \begin {array} {ccc} u_j & -a d y_j & -a d x_j - b d y_j \\ x_j & u_j - b x_j -c y_j & - c x_j - d y_j \\ y_j & a x_j & u_j - c y_j  \\\end {array} \right) .$$ Then define $$N_1 N_2 = N_3. $$ Taking determinants gives such an identity. 
