It is well known that the Lie group $Spin(9)$ acts on the vector space $\mathbb{R}^{16}$ (see e.g. Harvey's book "Spinors and calibrations".) It is convenient to identify this vector space with the octonionic plane $\mathbb{R}^{16}\simeq \mathbb{O}^2$. 


**Question. With this identifications, I would like to write down generators of the Lie algebra $spin(9)$ as explicitly as possible.**


A reference would be helpful.



Explicit generators of the Lie algebra $spin(9)$
There are various places where you can see this written down, but let me suggest some notes that I wrote about spinors in the low dimensions that includes what you want, assuming that you know something about the octonions. 


Here is the reference [Spinors in the low dimensions](https://services.math.duke.edu/~bryant/Spinors.pdf)


The formula that you want is at the bottom of Page 4 of these notes.


