I'll confine G as a matrix lie group.

Think of tangent vectors as "velocity vectors" of points moving smoothly through the point 1.

If , then . Let M=M(t) be a smooth path originating at 1. Apply d/dt to . Then, we have

.

Since , if we let , then we have

for t=0. Further, for any vector X with , the matrix is in (verify this). Thus, the tangent space of SO(n) consists of real vectors X such that .

You can apply this method for U(n), SP(n) as well.