Let $\{a,b,c\}$ be a set of simple roots of the Lie algebra $B_3$ and suppose $|a|=|c|$ and $\langle b,c\rangle=0$. I want to find the corresponding Cartan matrix.
I know that it's a $3\times3$ matrix with diagonal elements equal to $2$. Also since $\langle b,c\rangle=0$ then entry $(2,3)$ in the Cartan matrix is $\langle b^\vee,c\rangle=\frac{2}{\langle b,b\rangle}\langle b,c\rangle=0$ and so entry $(3,2)$ is also zero.
So far the Cartan matrix is $\begin{pmatrix} 2 & a_{12} & a_{13} \\ a_{21} & 2 & 0 \\ a_{31} & 0 & 2 \end{pmatrix}$. The determinant is $-2a_{21}a_{12}-2a_{31}a_{13}+8$, which should be strictly positive.
Since $|a|=|c|$ then $a_{13}=a_{31}$ by symmetry and so we must have that $a_{13}=a_{31}\in\{0,-1\}$. If $a_{31}=a_{13}=0$ then we don't have much more info about $a_{12}$ and $a_{21}$. If $a_{31}=a_{13}=-1$, then we know that none of $a_{12}$ or $a_{21}$ cannot be $-3$, or else it would violate that the determinant should be strictly positve.
So this is where I'm stuck. I haven't really managed to get any further.
Here $b$ is the small simple root, $a,c$ the long simple roots. Hence the Cartan matrix will be $\begin{pmatrix} 2 &-2 &-1 \\ -1& 2& 0 \\ -1& 0& 2 \end{pmatrix}$