I have a question regarding the inner product that appears in the study of root systems in Lie theory.
I have not seen anywhere an algorithm that explicitly calculates the product $(\alpha_i^\vee,\beta)$ appearing in the formula $$s_i(\beta)=\beta-(\alpha_i^\vee,\beta) \alpha_i,$$ when $\beta$ is arbitrary. I know there are some special formulas when $\beta$ is a simple root or a fundamental weight. But, in general, I have not seen any effective method, at least in the resources I have. I saw this example somewhere, which is of type $B$: $s_3(s_2s_1s_3s_2(\varpi_2)) = s_2s_1s_3s_2(\varpi_2) -2 \alpha_3$.
Can anyone explain how this is gotten? Sorry if the question is too elementary!
I have decided that in addition to my comments it might be helpful to just see some of the calculations. Firstly, I assume that we are in $B_n$ for $n\geq 4$ so these are all long roots we are dealing with. Then we can see that with $i,j \in \{1,2,3\}$ we have $ (\alpha_i^\vee,\alpha_i) = 2$, $(\alpha_i^\vee,\alpha_j) = -1$ for $j = \pm 1$ and $(\alpha_i^\vee,\alpha_j) = 0$ otherwise. Additionally $(\alpha_i^\vee,\varpi_j) = 0$ for $i \neq j$ and $(\alpha_i^\vee,\varpi_i) = 1$. We can read these off a Cartan matrix or note that $(\alpha_i^\vee,\alpha_j)(\alpha_j^\vee,\alpha_i)$ is exactly the number of lines between the $i$ and $j$ nodes on the Dynkin diagram by definition.
So we can then compute the repeated reflections by hand and linearity of $(\cdot,\cdot)$: $$ s_2(\varpi_2) = \varpi_2 - (\alpha_2^\vee,\varpi_2)\alpha_2= \varpi_2 - \alpha_2$$ $$ s_3s_2(\varpi_2)= \varpi_2 - \alpha_2 - (\alpha_3^\vee,\varpi_2 - \alpha_2)\alpha_3= \varpi_2 - \alpha_2 - \alpha_3$$ $$ s_1s_3s_2(\varpi_2)= \varpi_2 - \alpha_2 - \alpha_3 - \alpha_1$$ $$ s_2s_1s_3s_2(\varpi_2)= \varpi_2 - 2\alpha_2 - \alpha_3 - \alpha_1 $$ $$ s_3s_2s_1s_3s_2(\varpi_2)= \varpi_2 - 2\alpha_2 - \alpha_3 - \alpha_1 $$
Note the last two are the same which contradicts what your example says and implies that $s_2s_1s_3s_2(\varpi_2)$ is in fact orthogonal to $\alpha_3$ so is unmoved by the final $s_3$ reflection. I double checked in LiE that the last two stages are the same so that should be correct.