Consider a semisimple algebraic group $G$ over an algebraically closed field of arbitrary characteristic and let $\bar U(G)$ denote its hyperalgebra (ie, the restricted Hopf dual of the coordinate ring of $G$). In characteristic 0, $\bar U(G)$ is just the enveloping algebra of Lie($G$) but this is not the case in positive characteristic. There is a standard degree filtration $\bar U_{\leq n}(G)$ on $\bar U(G)$. For an element $X \in \bar U(G)$ let us define the degree $\textrm{deg}(X)$ of $X$ to be the minimum $n$ such that $X \in \bar U_{\leq n}(G)$.

It is a straightforward verification that for $X$ of degree n and $Y$ of degree m we have $XY - YX \in \bar U_{\leq n + m -1}(G)$ (cf for example Jantzen's Representations of Algebraic Groups). What I am wondering is when we achieve the maximum degree; eg, when is it the case that $\textrm{deg}(XY - YX) = n + m -1$? I would not be surprised if this always holds in characteristic 0, ie when we are just considering the enveloping algebra. However, it is clear that this equality does not always hold in positive characteristic since there are zero divisors in $\bar U(G)$ in that case. Nevertheless, I would be happy if there was a weaker statement that could be made, perhaps something like: $\textrm{deg}(XY - YX) = n + m -1$ when $X,Y$ are basis element monomials (with respect to some ordering of Lie($G$)) such that $XY \neq 0$.

Consider a semisimple algebraic group $G$ over an algebraically closed field of arbitrary characteristic and let $\bar U(G)$ denote its hyperalgebra (ie, the restricted Hopf dual of the coordinate ring of $G$). In characteristic 0, $\bar U(G)$ is just the enveloping algebra of Lie($G$) but this is not the case in positive characteristic. There is a standard degree filtration $\bar U_{\leq n}(G)$ on $\bar U(G)$. For an element $X \in \bar U(G)$ let us define the degree $\textrm{deg}(X)$ of $X$ to be the minimum $n$ such that $X \in \bar U_{\leq n}(G)$.

It is a straightforward verification that for $X$ of degree n and $Y$ of degree m we have $XY - YX \in \bar U_{\leq n + m -1}(G)$ (cf for example Jantzen's Representations of Algebraic Groups). What I am wondering is when we achieve the maximum degree; eg, when is it the case that $\textrm{deg}(XY - YX) = n + m -1$?

I would not be surprised if this always holds in characteristic 0, ie when we are just considering the enveloping algebra. However, it is clear that this equality does not always hold in positive characteristic since there are zero divisors in $\bar U(G)$ in that case. Nevertheless, I would be happy if there was a weaker statement that could be made, perhaps something like: $\textrm{deg}(XY - YX) = n + m -1$ when $X,Y$ are basis element monomials (with respect to some ordering of Lie($G$)) such that $XY \neq 0$.

EDIT: As Bruce Westbury points out below, a lot of what I wrote above is silly and I've struck it out. Perhaps my question still has merit though.