If V is a 3-dimensional Lie algebra with basis vectors E,F,G with Lie bracket relations [E,F]=G, [E,G]=0, [F,G]=0 and V' is the Lie algebra consisting of all 3x3 strictly upper triangular matrices with complex entries then would you say the following 2 mappings (isomorphisms) are different? I had to give an example of 2 different isomorphisms between these vector spaces.
It's been a while since you posted and nobody's replied yet...maybe you should check out MarkedbyTeachers.com, TSR's sister site. It has the largest library of essays in the UK.
They've got over 181,000+ coursework, essays, homeworks etc.. all written by GCSE, A Level, University and IB students across all topics. You get access either by publishing some of your own work, or paying £4.99 for a month's access. Both ways give you unlimited access to all of the essays.
All their documents are submitted to Turnitin anti-plagiarism software, so it can't be misused, and the site's used by hundreds of thousands of UK teachers and students.
Maybe this post is a little late given that the thread is a week old now, but recall that Lie algebra isomorphisms are also linear transformations, so are determined by their values on the basis elements E,F and G. So it seems your two maps are the same. However, the map you've given is an isomorphism and from this, it doesn't look too difficult to find a second?