While it is easy to find many sources that give expressions for the (co)roots of an abstract root system, it is less easy to find a reference that gives explicit matrices that are the "coroots" (in the sense of Lie algebra elements) in the simple classical Lie algebras (i.e. types $A_n$, $B_n$, $C_n$ and $D_n$). I'm particularly interested in those corresponding to the long roots, but a reference that gave all of them would be perfect. **Edit:** to clarify, I mean the compact forms of the real matrix Lie algebras.

The reason I ask is that it is useful to normalise the Killing form so that the these algebra elements have length $\sqrt{2}$, and I want to give these normalisations as examples for the classical Lie algebras for a paper I'm writing, since sources I'm familiar with just specify the case $\mathfrak{su}(n)$. For the non-simply-laced cases it is less easy to sort out what's going on, as my background in Lie theory is weak, so chasing the definitions from the abstract root system through the corresponding decomposition of the Lie algebra etc is not obvious. But a source that just gives the answer is not forthcoming after a decent internet search!

## Best Answer

This is in Fulton and Harris. Look on page 240-1 for $\mathfrak{sp}$ and page 270-1 for $\mathfrak{so}$.

EDIT:Apparently the issue here is that the OP wants to change basis to the standard version of the bilinear forms. For $\mathfrak{sp}_{2n}$, there's no issue here. All real symplectic forms are equivalent. If you want to think about the compact form as transformations of $\mathbb{H}^n$ preserving quaternionic norm, Fulton and Harris are thinking about this as a complex vector space with basis $e_1,\dots, e_n, je_1,\dots, je_n$.More explicitly, the Lie algebra is the anti-Hermitian quaternionic $n\times n$ matrices. The "obvious" Cartan is given by diagonal matrices with values in $i\mathbb{R}$ (note: nothing special about $i$; conjugating by unit quaternions sends this to real multiples of any imaginary quaternion), and the SU(2)'s for the simple roots are given by:

For $\mathfrak{so}_{n}$, you need to change the real form to pair together the basis vectors that pair non-trivially. The torus of $\mathfrak{so}_{n}$ acts on these real 2-d spaces by the usual rotation. That is, they are block diagonal with $2\times 2$-blocks given by $$\begin{bmatrix}\cos \theta_i & \sin \theta_i\\ -\sin \theta_i & \cos \theta_i \end{bmatrix}$$ The root SU(2)'s come from looking at a $4\times 4$ block along the diagonal, and picking out one of the factors of $SU(2) x SU(2) =Spin(4)$ with the exception of:

EDIT AGAIN:. I like a good challenge, though this is all getting a little complicated (you have read the twitter threads too if you want all the details). Coroot vectors in the Lie algebra of the compact group don't make sense, so I think it's better to think about homomorphisms of $\mathfrak{su}(2)$ into your Lie algebra. You can think of any unit imaginary quaternion in $\mathfrak{su}(2)$ (for example, any of the Pauli spin matrices) as a coroot vector if you want; the norm-square of the coroot under your form is the pairing of this vector with itself (up to sign). As I said on twitter, there are just a few basic building blocks of these, so let me write those out. Let $X$ be an imaginary quaternion:It is an assignment to you to write these in your preferred notation.

rightmultiplication by $X$ on the quaternions as a diagonal block.