  We start with the Lorentz algebra $ L=o_{R}(1,3)$ over the reals and find a
suitable basis $B$ relative to which the structure constants are integers. Thus
we consider the $Z$-algebra $L_{Z}$ which is free as a $Z$-module and its
$Z$-basis is $B$. This allows us to define the Lorentz type algebra
$L_K:=L_{Z}\otimes_{Z} K$ over any field $K$. In a similar way, we consider
Poincar\'e type algebras over any field $K$. In this paper we study the ideal
structure of Lorentz and of Poincar\'e type algebras over different fields. It
turns out that Lorentz type algebras are simple if and only if the ground field
has no square root of $-1$. Thus, they are simple over the reals but not over
the complex. Also, if the ground field is of characteristic $2$ then Lorentz
and Poincar\'e type algebras are neither simple nor semisimple. We extend the
study of simplicity of the Lorentz algebra to the case of a ring of scalars
where we have to use the notion of $m$-simplicity (relative to a maximal ideal
$m$ of the ground ring of scalars). The Lorentz type algebras over a finite
field $F_q$ where $q=p^n$ and $p$ is odd are simple if and only if $n$ is odd
and $p$ of the form $p=4k+3$. In case $p=2$ then the Lorentz type algebra are
not simple. Once we know the ideal structure of the algebras, we get some
information of their automorphism groups. For the Lorentz type algebras (except
in the case of characteristic $2$) we describe the affine group scheme of
automorphisms and the derivation algebras. For the Poincar\'e algebras we
restrict this program to the case of an algebraically closed field of
characteristic other than $2$.
