# Whitehead lemma for exceptional Chevalley groups

Let $R$ be a commutative ring with $1$. Let $u$ be a column and $v$ be a row of length $n$ over $R$, such that $vu=0$. Then the matrix $1+uv$ belongs to $\mathrm{SL}(n,R)$ and is called linear transvection. Whitehead lemma asserts that $1+uv$ belongs to $\mathrm{E}(n+1,R)$, where $\mathrm{E}(n+1,R)$ denotes the elementary subgroup of $\mathrm{SL}(n+1,R)$. More precisely $$$$\begin{pmatrix} 1+uv & 0\\ 0 & 1\end{pmatrix}=\left[\begin{pmatrix} 1 & u\\ 0 & 1\end{pmatrix},\begin{pmatrix} 1 & 0\\ v & 1\end{pmatrix} \right],\tag{1}$$$$ where $[g,h]=ghg^{-1}h^{-1}$ denotes the commutator of elements $g$ and $h$. There are variations of this lemma for other classical Chevalley groups and hyperbolic unitary group [see Bak, Vavilov "Structure of Hyperbolic Unitary Groups I: Elementary Subgroups", section 7].

Are there variations of this lemma for exceptional Chevalley groups? Consider, for example, embedding $\mathsf D_5\subseteq \mathsf E_6$ and let $\varphi\colon\mathrm G(\mathsf D_5,R)\to \mathrm G(\mathsf E_6,R)$ be the homomorphism of simply connected Chevalley groups induced by this embedding. Simply connected Chevalley group of type $\mathsf D_5$ is the spin group $\mathrm{Spin}(10,R)$. Elements of the spin group, corresponding to linear transvections, are of the form $1+u\cdot v$, where $u$ is an isotropic vector and $v$ is a vector, such that $(u,v)=0$ [see Hahn, O'Meara "The Classical Groups and K-Theory", section 7.2A]. Is it true that the image of $1+u\cdot v$ under $\varphi$ belongs to the elementary subgroup $\mathrm{E}(\mathsf E_6,R)$? Is there formulae like $(1)$ in this case?