  We systematically investigate axisymmetric extremal isolated horizons (EIHs)
defined by vanishing surface gravity, corresponding to zero temperature. In the
first part, using the Newman-Penrose and GHP formalism we derive the most
general metric function for such EIHs in the Einstein-Maxwell theory, which
complements the previous result of Lewandowski and Pawlowski. We prove that it
depends on 5 independent parameters, namely deficit angles on the north and
south poles of a spherical-like section of the horizon, its radius (area), and
total electric and magnetic charges of the black hole. The deficit angles and
both charges can be separately set to zero. In the second part of our paper, we
identify this general axially symmetric solution for EIH with extremal horizons
in exact electrovacuum Plebanski-Demianski spacetimes, using the convenient
parametrization of this family by Griffiths and Podolsky. They represent all
(double aligned) black holes of algebraic type D without a cosmological
constant. Apart from a conicity, they depend on 6 physical parameters (mass,
Kerr-like rotation, NUT parameter, acceleration, electric and magnetic charges)
constrained by the extremality condition. We were able to determine their
relation to the EIH geometrical parameters. This explicit identification of
type D extremal black holes with a unique form of EIH includes several
interesting subclasses, such as accelerating extremely charged
Reissner-Nordstrom black hole (C-metric), extremal accelerating Kerr-Newman,
accelerating Kerr-NUT, or non-accelerating Kerr-Newman-NUT black holes.
