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We study `good elements' in finite $2n$-dimensional classical groups $G$:namely $t$ is a `good element' if $o(t)$ is divisible by a primitive primedivisor of $q^n-1$ for the relevant field order $q$, and $t$ fixes pointwise an$n$-space. The group ${\rm{SL}}_{2n}(q)$ contains such elements, and they arepresent in ${\rm{Su}}_{2n}(q), {\rm{Sp}}_{2n}(q), {\rm{So}}^\epsilon_{2n}(q)$,only if $n$ is odd, even, even, respectively. We prove that there is anabsolute positive constant $c$ such that two random conjugates of $t$ generate$G$ with probability at least $c$, if $G\ne {\rm{Sp}}_{2n}(q)$ with $q$ even.In the exceptional case $G={\rm{Sp}}_{2n}(q)$ with $q$ even, two conjugates of$t$ never generate $G$: in this case we prove that two random conjugates of $t$generate a subgroup ${\rm{SO}}^\epsilon_{2n}(q)$ with probability at least $c$.The results (proved for all field orders at least $4$) underpin analysis of newconstructive recognition algorithms for classical groups in evencharacteristic, which succeed where methods utilising involution centralisersare not available.

Generation of finite classical groups by pairs of elements with large fixed point spaces