  The purpose of this paper is twofold. One is to enrich from a geometrical
point of view the theory of octonionic slice regular functions. We first prove
a boundary Schwarz lemma for slice regular self-mappings of the open unit ball
of the octonionic space. As applications, we obtain two Landau-Toeplitz type
theorems for slice regular functions with respect to regular diameter and slice
diameter respectively, together with a Cauchy type estimate. Along with these
results, we introduce some new and useful ideas, which also allow to prove the
minimum principle and one version of the open mapping theorem. Another is to
strengthen a version of boundary Schwarz lemma first proved in \cite{WR} for
quaternionic slice regular functions, with a completely new approach. Our
quaternionic boundary Schwarz lemma with optimal estimate improves considerably
a well-known Osserman type estimate and provides additionally all the extremal
functions.
