According to Conway and Sloane's, Sphere Packings, Lattices and Groups, we can represent a lattice by its Gram Matrix. Such a matrix $A=MM^T$ where $M$ is the matrix consisting of the basis vectors for the lattice, $\Lambda$.
Furthermore, the automorphism group of $\Lambda$, denoted $Aut(\Lambda)$ is described as "the set of all distance preserving transformations of the space such that the origin is fixed and the lattice is mapped to itself."
My slew of questions:
1) What do they mean by distance preserving?
2) Does mapping the lattice to itself mean that each basis vector is mapped to another basis vector?
3) Could someone provide simple examples and display some elements of $Aut(\Lambda)$ as orthogonal matrices?
You are missing some detail. In any case I would not begin with SPLAG. I like Rational Quadratic Forms by Cassels. If your main interest is getting ready for SPLAG, the two items I recommend are From Error Correcting Codes Through Sphere Packings to Simple Groups by T. M.Thompson, also Lattices and Codes by W. Ebeling. Conway also does beginning lattices in The Sensual Quadratic Form. What you have is a square symmetric matrix, positive definite. If written as the Hessian matrix of a quadratic form, it has all integer entries. So call it $H.$
An automorphism is a square matrix $P$ of determinant $1$ such that $$ P^T H P = H. $$ These need not be orthogonal in the general matrix sense.
In the 2 by 2 case, these groups are quite small, often just the identity and its negative.
I wrote programs for the 3 by 3 case, let me give an example....
I'm not sure whether this program restricts to positive determinant, it might or might not. If restricted, there would be 24 matrices $P,$ otherwise $48.$