  We study simple Lie algebras generated by extremal elements, over arbitrary
fields of arbitrary characteristic. We show: (1) If the extremal geometry
contains lines, then the Lie algebra admits a $5 \times 5$-grading that can be
parametrized by a cubic norm structure; (2) If there exists a field extension
of degree at most $2$ such that the extremal geometry over that field extension
contains lines, and in addition, there exist symplectic pairs of extremal
elements, then the Lie algebra admits a $5 \times 5$-grading that can be
parametrized by a quadrangular algebra.
  One of our key tools is a new definition of exponential maps that makes sense
even over fields of characteristic $2$ and $3$, which ought to be interesting
in its own right.
