let $f$ be a field and $n>0$ a positive integer. let $a = a_0 + a_1 + ...+ a_n$ be the exterior algebra of dimension $2^n$ over $f$ with its natural grading. then a homogenous element w in $a_s$ generates a homogeneous principal ideal $wa$. what is the maximum value of $\dim_f(wa \cap a_r)$ for given $s,r,n$ as $w$ varies in $a_s$? we state a conjecture. the (most interesting?) case $(s,r,n) = (3,6,9)$ (where the max is 84) is directly related to the exceptional lie algebra $e_8$. (by definition, a is generated by elements $e_1,...,e_n$ that satisfy $(1) e_i^2 =0$ and $(2) e_ie_j +e_je_i=0$ for $1<=i,j<= n$. the generators $e_i$ all lie in $a_1$. by definition, $a_s$ is the $f-span$ of all $s-fold$ products $e_{j_1} ... e_{j_s}.)$