An explicit example that is my favourite is below which is in Humphreys bookon Linear Algebraic Groups.

For a field $k$, let $T(n,k)$ denote the group of $n\times n$ non-singular upper triangular group and $D(n,k)$ the non-singular diagonal matrices and $U(n,k)$ upper triangular matrices with 1's in the diagonal.

$1\to U(n,k)\to T(n,k)\to D(n,k)\to 1$ is actually a split-sequence. This can be generalized to any connected solvable group leading to its structure theorem as semi-direct product of maximal unipotent normal subgroup and maximal torus subgroup.

An explicit example that is my favourite is below which is in Humphreys book on Linear Algebraic Groups.

For a field $k$, let $T(n,k)$ denote the group of $n\times n$ non-singular upper triangular group and $D(n,k)$ the non-singular diagonal matrices and $U(n,k)$ upper triangular matrices with 1's in the diagonal.

$1\to U(n,k)\to T(n,k)\to D(n,k)\to 1$ is actually a split-sequence. This can be generalized to any connected solvable group leading to its structure theorem as semi-direct product of maximal unipotent normal subgroup and maximal torus subgroup.