I’m having a problem understanding Cartan’s notation in theory of spinors §2. Here are the definitions:
Set x to $3 \\ 4$ so the length is 5, then set as a simple test basis $\eta_1$ to $1 \\ 1$ and $\eta_2$ to $0 \\ 1$.
$u^1$ I set to 3 and $u^2$ to 1. The scalar product of $\eta_1 . \eta_2$ is 1. If I run over the $u^i u^j$ successively and independently, that gives me $$3 \quad 3 \\ 3 \quad 1 \\ 1 \quad 3 \\ 1 \quad 1$$, which sum to 16, not 25.
If changing basis is supposed to change length, then I don’t understand why Cartan’s $$u^i u^j \eta_1 . \eta_2$$ is a good definition of fundamental form.
Let $x = (3, 4) = 3e_1 + 4 e_2$, with square length $3^2 + 4^2 = 25$. Now express $x$ in the new basis of $\eta_1 = (1, 1)$ and $\eta_2 = (0, 1)$, and we get that $$ x = (3, 4) = (3, 3) + (0, 1) = 3 \eta_1 + \eta_2.$$ Therefore in the expression $x = u^i\eta_i$ we have $u^1 = 3$ and $u^2 = 1$, as you found.
Now consider the fundamental form $g_{ij} = \eta_i \cdot \eta_j$, which we can write as a matrix $$ g = \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & 1\end{pmatrix}.$$ Evaluating $\Phi$ we get $$ \begin{aligned} \Phi &= g_{ij} u^i u^j\\ &= g_{11} u^1 u^1 + g_{12} u^1 u^2 + g_{21} u^2 u^1 + g_{22} u^2 u^2 \\ &= 2 \cdot 3 \cdot 3 + 1 \cdot 3 \cdot 1 + 1 \cdot 1 \cdot 3 + 1 \cdot 1 \cdot 1 \\ &= 25. \end{aligned} $$