Analogue of $(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2$ for vectors The Brahmagupta–Fibonacci identity $(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2$ allows us to write a product of squares as a sum of squares. Is there an analogue of this identity when $a, b, c, d$ are vectors in $\mathbb{R}^n$ and the multiplication is the ordinary scalar product?
One of the main hurdles is of course $(a \cdot c)(b \cdot d) \neq (a \cdot b)(c \cdot d)$. The Cauchy-Binet identity gives an expression for  $(a \cdot c)(b \cdot d) - (a \cdot b)(c \cdot d)$, however I do not see a way to turn this into a simple expression.
EDIT: I do not mind having extra terms in the end: something like $(a\cdot a + b \cdot b)(c \cdot c + d \cdot) = (a\cdot c - b\cdot d)^2 + (a\cdot d + b\cdot c)^2 + \mbox{other terms}$ where the remaining terms can be written explicitly.

I assume (perhaps unreasonably closed-mindedly) that you want an analogue of the form
\begin{align}
\left(\left<a,a\right>^2 + \left<b,b\right>^2\right) \left(\left<c,c\right>^2 + \left<d,d\right>^2\right) = \left<v,v\right>^2 + \left<w,w\right>^2
\end{align}
that holds for any vectors $a, b, c, d \in \mathbb{Q}^n$, where $v$ and $w$ are two vectors in $\mathbb{Q}^n$ whose entries are fixed degree-$2$ homogeneous polynomials in the entries of $a, b, c, d$. Such an analogue exists only if $n \in \left\{1,2,4\right\}$. Indeed, if we number the entries of the vectors $a, b, c, d$ such that $a = \left(x_1, x_2, \ldots, x_n\right)^T$ and $b = \left(x_{n+1}, x_{n+2}, \ldots, x_{2n}\right)^T$ and $c = \left(y_1, y_2, \ldots, y_n\right)^T$ and $d = \left(y_{n+1}, y_{n+2}, \ldots, y_{2n}\right)^T$, then the above analogue would imply that the polynomial $\left(x_1^2 + x_2^2 + \cdots + x_{2n}^2\right) \left(y_1^2 + y_2^2 + \cdots + y_{2n}^2\right)$ can be written as $u_1^2 + u_2^2 + \cdots + u_{2n}^2$ for some quadratic forms $u_1, u_2, \ldots, u_{2n}$ in the $x_i$ and $y_i$. But such quadratic forms only exist if $n \in \left\{1,2,4\right\}$, according to Theorem 1.1 in Keith Conrad, The Hurwitz theorem on sums of squares by representation theory. (Note that my $2n$ is his $n$.)
