Let $R$ be a reduced, irreducible, crystallographic root system with positive roots $R^+$ and simple roots $\Delta$. Let $W$ be the Weyl group of $R$. Let $(-,-)$ be a $W$-invariant scalar product on $\mathbb{R}\Delta$ (this is unique up to non-zero scalar as far as I know). 


**Question 1**


Let $\alpha,\beta\in R^+$ such that $(\alpha,\beta)\geq 0$ and such that $\alpha+\beta\in R$. Is then always $(\alpha+\beta)^\vee=\alpha^\vee+\beta^\vee$?


Here we denote as usual by $\alpha^\vee=2\alpha/(\alpha,\alpha)$ the dual root / coroot of a root $\alpha$.


**Question 2**


Let $\alpha,\beta\in R^+$ such that $(\alpha,\beta)=0$ and such that $\alpha+\beta\in R$. Is then always $(\alpha+\beta)^\vee=\alpha^\vee+\beta^\vee$?


The reason why I cannot answer this question myself is that I am not familiar with pairs of roots which are orthogonal but are not strongly orthogonal. We say $\alpha,\beta\in R$ are strongly orthogonal if $\alpha\pm\beta\notin R\cup\{0\}$.


I understand that such pairs of roots can only occur if there are two root length and if
$$
(\text{long root length})^2=2\cdot(\text{short root length})^2\,,
$$
i.e. only in type $\mathsf{BCF}$. But still I am not able to find them. I cannot find roots for which the hypotheses are satisfied.


NB. It also counts as an answer if you give some examples of such pairs of roots and verify **Question 2** on them. In any case, please use the numbering of the simple roots as in the Bourbaki tables. It makes it much easier to follow for me.


Thank you very much for any kind of help!



On pairs of roots which are orthogonal but not strongly orthogonal
**Lemma**


Let $\alpha,\beta\in R^+$ such that $(\alpha,\beta)\geq 0$ and such that $\alpha+\beta\in R$. Then we have $(\alpha+\beta)^\vee<\alpha^\vee+\beta^\vee$.


**Proof**


Sine $(\alpha,\beta)\geq 0$, we see that there are two root length, and that $\alpha+\beta$ is long and $\alpha,\beta$ are short. Let
$$
n=\frac{(\text{long root length})^2}{(\text{short root length})^2}\,.
$$
Then $n$ is a strictly positive integer, more concretely $n=2,3$. A simple computation shows that $\alpha^\vee+\beta^\vee=n(\alpha+\beta)^\vee$, in particular $(\alpha+\beta)^\vee<\alpha^\vee+\beta^\vee$.


**Example**


Let $R$ be of type $\mathsf{B}\_2$. Then $\alpha=\alpha\_1+\alpha\_2$ and $\beta=\alpha\_2$ are a pair of orthogonal roots which are not strongly orthogonal. Here we have $n=2$ and $\alpha^\vee+\beta^\vee=2(\alpha+\beta)^\vee=2(\alpha\_1^\vee+\alpha\_2^\vee)$.


