I was using the built-in functions for Root Systems in SAGE, and I noticed that the Cartan Matrices for Type $B_n$ and type $C_n$ are interchanged from what I thought they would be, i.e. following the Plates in the back of Bourbaki's Lie Groups and Lie Algebras, vol. 4-6.
Are there different conventions for choosing simple roots from these two root systems? I figured that everyone followed Bourbaki in this regard, but would like to hear about any competing conventions.
This question (which I overlooked for a long time) reflects a natural notational confusion but is easy to answer. The Cartan integers themselves are unambiguous for each root system, but the meaning of the two indices used in writing $c_{i,j}$ is conventional and is reversed in some sources. For types $B,C$ that reversal leads to transposed matrices, which is what you are seeing. Bourbaki and most other sources now write $c_{i,j} := 2(\alpha_i, \alpha_j)/(\alpha_j, \alpha_j)$ relative to a symmetric bilinear form which historically derives from the Killing form; but sometimes this is instead wrutten $c_{j,i}$. For better or worse, there is no secret supercommittee dictating a single choice.
The labels $A, B, \dots$ for the simple Lie algebras or associated root systems are of course conventional too (going back to Killing and Cartan) but have become standard by now in the literature. The subscript (often $\ell$) indicating the Lie algebra rank is conventionally restricted in an arbitrary way to avoid assigning multiple labels to isomorphic Lie algebras such as $B_2, C_2$. (You could decide to write $C_1$ and discard $A_1$, but hardly anyone does this.)
Above rank 2 the series $C_\ell$ is assigned to the Lie algebras of symplectic groups, where there is a unique long simple root. The series $B_\ell$ belongs to Lie algebras of special orthogonal groups in odd dimensions, where there is a unique short simple root. Numbering the simple roots in a Dynkin diagram is again purely conventional (and not always the same in textbooks), but usually the last simple root $\alpha_\ell$ in these cases is the unique one of its length.
Lie theory has developed in multiple directions, leading to many differences in notation and terminology; but fortunately everyone tends to agree on the underlying ideas.
