TITLE: A general pattern to find the roots of the classical lie algebras
QUESTION [6 upvotes]: Is there any general pattern for the roots of each of the classical lie algebras? So, can I tell all the roots of each of the $nth$ rank classical lie algebras $A_n, B_n, C_n, D_n$, as a linear combination of the simple roots. So I am looking for a pattern for the roots at each level given a classical lie algebra. So, if I am given $C_6$ for e.g., is there any general formula for the roots at say the $kth$ level in terms of $k$, e.g. $f(k,n)a_1+g(k,n)a_2+...$ are the roots, where $a_1,a_2,...$ are the simple roots.

REPLY [3 votes]: A summary:
For any root $\alpha$, the set of basis vectors $\alpha_i$ occuring in the expansion of $\alpha$ (i.e. the ones with non-zero coefficients) must form a connected set of nodes
in the Dynkin diagram. For types $A,B,C,D$ only 0, 1 and 2 occur as coefficients. The coefficient 2 is forbidden in type $A$.
For type $A_n$ the sequences of coefficients are of the form $0^a1^{1+b}0^c$. Here $0^a$ denotes a string (possibly empty) of $a$ zeros, and $1^b$ a string of $b$ ones. Obviously $a,b,c$ are all non-negative integers subject to the constraint $a+b+c=n-1$.
For type $B_n$, in addition to the sequences that occurred in type $A_n$, also sequences of the form $0^a1^{1+b}2^{1+c}$ with $a+b+c=n-2$ are ok. Observe that there has to be at least a single 1 between the zeros and the 2s. The 2s are at the end of the lone short basic root (check your source: conceivably some may have the short basic root at the other end).
For type $C_n$, in addition to the sequences that occurred in type $A_n$, the sequences of the form $0^a1^b2^{1+c}1$ are ok. Again $a+b+c=n-2$. The string of 2s thus does not include, but is connected to, the lone long (rem. this was wrongly written as "short" in the earlier version) basic root.
For type $D_n$, any combination of 0s and 1s such that the 1s form a connected set in the Dynkin diagram is ok. In addition to that, sequences of the form
$0^a1^{1+b}2^c\vphantom{(}^{\displaystyle{1}}_{\displaystyle{1}}$ with $a+b+c=n-3$ are ok. IOW the string of 2s must end at the vertex connected to three other vertices, and it must also have a "buffer" of at least a single 1 in each three directions separating the 2s from 0s.