New form of all black holes of type D with a cosmological constant

We present an improved metric form of the complete family exact black hole spacetimes algebraic type D, including any cosmological constant. This class was found by Debever in 1971, Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski and Demia\ifmmode 1976, conveniently reformulated Griffiths Podolsk\'y 2005. In our new this key functions are simplified, partially factorized, fully explicit. They depend on seven parameters with direct physical meanings, namely $m,a,l,\ensuremath{\alpha},e,g,\mathrm{\ensuremath{\Lambda}}$ which characterize mass, Kerr-like rotation, NUT parameter, acceleration, electric magnetic charges hole, constant, respectively. Moreover, general reduces directly to familiar forms (possibly accelerating) Kerr-Newman--(anti--)de Sitter spacetime, charged Taub-NUT--(anti--)de solution, or rotating charged) $C$-metric a constant simply setting corresponding zero. addition, it shows that \'{n}\fi{}ski-Demia\ifmmode does not involve accelerating holes without rotation. The also enables us study various geometrical properties, character singularities, two cosmo-acceleration horizons (in generic situation), related ergoregions, global structure Penrose conformal diagrams, cosmic strings causing acceleration holes, their pathological regions closed timelike curves, thermodynamic quantities.