  To each totally disconnected, locally compact topological group G and each
group A of automorphisms of G, a pseudo-metric space of ``directions'' has been
associated by U. Baumgartner and the second author. Given a Lie group G over a
local field, it is a natural idea to try to define a map from the space of
directions of analytic automorphisms of G to the space of directions of
automorphisms of the Lie algebra L(G) of G, which takes the direction of an
analytic automorphism of G to the direction of the associated Lie algebra
automorphism. We show that, in general, this map is not well-defined. However,
the pathology cannot occur for a large class of linear algebraic groups (called
``generalized Cayley groups'' here). For such groups, the assignment just
proposed defines a well-defined isometric embedding from the space of
directions of inner automorphisms of G to the space of directions of
automorphisms of L(G). Some counterexamples concerning the existence of small
joint tidy subgroups for flat groups of automorphisms are also provided.
